Изменения

Таблица умножения(таблица Кэли) — таблица, которая описывает структуру конечных алгебраических систем с одной бинарной операцией. Таблица позволяет определить, является ли группа абелевой, найти ядро группы и обратные элементы по отношению к другим элементам в этой группе.

 '''=== Структура'''===Пусть <mathtex>\mathbb{A}_n</math> = <math>\{a_1,a_2,\dots,a_n\}</mathtex> - — группа из <tex>n </tex> элементов.

'''=== Свойства'''==={{Утверждение|statement=Каждая строка или столбец являются перестановкой элементов группы.|proof=Пусть <tex>a,b,c,d \in G</tex>. Тогда <tex>ab=d</tex> и <tex>ac=d \Rightarrow b=c</tex>. Так как количество клеток в строке равно количеству элементов, то, по принципу Дирихле, каждый элемент группы встречается в строке один раз. }}{{Утверждение|statement=Если таблица симметрична относительно главной диагонали, то операция умножения коммутативна.|proof=Таблица симметрична <tex>\Rightarrow ab = ba</tex> для любых <tex>a,b \in G</tex>}}{{Утверждение|statement=В конечной группе порядок каждого элемента является делителем порядка группы.|proof=Рассмотрим элемент <tex>x\in G</tex> c порядком <tex>n</tex> и подмножество <tex>\langle x\rangle=\lbrace e,\,x,\,x^2,\,...,x^{n-1}\rbrace</tex> (все <tex>x^k</tex> различны при <tex>k<n</tex> — в противном случае при <tex>x^k=x^m (m<k<n)\Rightarrow x^{k—m}=e</tex>, т.е. <tex>n>k—m</tex> не является порядком элемента <tex>x</tex>). Легко проверить, что <tex>\langle x\rangle</tex> — подгруппа <tex>G</tex>. По [[Теорема Лагранжа|теореме Лагранжа]] порядок любой подгруппы делит порядок группы. Значит, и <tex>n</tex> делит порядок <tex>G</tex>.}}{{Утверждение|statement=Все группы простого порядка <tex>p</tex> изоморфны <tex>\mathbb{Z}/p\mathbb{Z}</tex>.|proof=Рассмотрим элемент <tex>x\in G,\,x\neq e</tex> c порядком <tex>n</tex> и подмножество <tex>\langle x\rangle=\lbrace e,\,x,\,x^2,\,...,x^{n-1}\rbrace</tex> (все <tex>x^k</tex> различны при <tex>k<n</tex> — см. выше). Очевидно, <tex>\langle x\rangle</tex> — подгруппа <tex>G</tex>, изоморфная <tex>\mathbb{Z}/n\mathbb{Z}</tex>. Но тогда <tex>n</tex> делит <tex>p</tex>(как порядок подгруппы) и не равняется единице(<tex>x^1\neq e</tex>), значит <tex>n=p</tex>. Раз порядок конечной подгруппы <tex>\mathbb{Z}/p\mathbb{Z}\subseteq G</tex> совпадает с порядком группы, то группа и подгруппа просто совпадают: <tex>\mathbb{Z}/p\mathbb{Z}\eqsim G</tex>.}}

1) Каждая строка или столбец являются перестановкой элементов === Примеры таблиц умножения для конечных групп ===Ниже перечислены все группыдо шестого порядка включительно:

2) Если таблица симметрична относительно главной диагонали, то операция умножения коммутативна  '''Построение''' Вследствие первого свойства, можно заполнить таблицу не имея всей информации об операции умножения. Если таблицу заполнить не удаётся(нарушается первое свойство - в ряде или колонке оказывается 2 одинаковых элемента), значит операции, удовлетворяющей данным соотношениям, не существует.  ''Алгоритм построения'': 1) заполнить "скелет" таблицы - ячейки в которых стоит нейтральный элемент. "Скелет" симметричен относительно главной диагонали(если a - обратный к b, то b - обратный к a, то есть любой элемент коммутирует со своим обратным). 2) используя известные соотношения и свойство 1 заполнить таблицу. ''Замечание'': по соглашению в заголовках таблицы * <tex>|G| = 1-ым идёт нейтральный элемент, затем элементы, которые совпадают с обратным, затем остальные. === Примеры таблиц умножения для конечных групп ===</tex>1) n = 1Тривиальная группа

2) n = 2{| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</bigtex> !style="background:#efefef;"| <big>a</big> G|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a2</bigtex>|-!style="backgroundГруппа вычетов по модулю два относительно сложения:#efefef;"| <bigtex>a</big>| <big>a</big> || <big>e</big>|} 3) n = 3\mathbb{| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big> !style="background:#efefef;"| <big>a</big> !style="background:#efefef;"| <big>b</big> |-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big>|-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>e</big>|-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>e</big> || <big>a</big>|Z} 4) n = 4{| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big> !style="background:#efefef;"| <big>a</big> !style="background:#efefef;"| <big>b</big> !style="background:#efefef;"| <big>c</big> |-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>|-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big>|-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>a</big>|-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>a</big> || <big>e</big>|} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big> !style="background:#efefef;"| <big>a</big> !style="background:#efefef;"| <big>b</big> !style="background:#efefef;"| <big>c</big> |-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>|-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>|-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>a</big>|-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>e</big> || <big>a</big> || <big>b</big>|} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big> !style="background:#efefef;"| <big>a</big> !style="background:#efefef;"| <big>b</big> !style="background:#efefef;"| <big>c</big> |-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>|-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big>|-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>a</big> || <big>e</big>|-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>e</big> || <big>a</big>|}5) n = 5{| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big> !style="background:#efefef;"| <big>a</big> !style="background:#efefef;"| <big>b</big> !style="background:#efefef;"| <big>c</big> !style="background:#efefef;"| <big>d</big> |-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big>|-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>e</big>|-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>d</big> || <big>e</big> || <big>a</big>|-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big>|-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big>|} 6) n = 6{| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>b</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>c</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>c</big> || <big>f</big> || <big>b</big> || <big>d</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>c</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>d</big> || <big>b</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>c</big> || <big>b</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>e</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>e</big> || <big>b</big> || <big>a</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big> |} 2\mathbb{| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>c</big> || <big>a</big> || <big>b</big> || <big>e</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>c</big> || <big>d</big> || <big>b</big> || <big>e</big> || <big>a</big> |Z} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</bigtex>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>c</big> || <big>d</big> || <big>b</big> || <big>a</big> || <big>e</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>c</big> || <big>e</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>c</big> || <big>d</big> || <big>b</big> || <big>a</big> || <big>e</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>e</big> || <big>d</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>e</big> || <big>a</big> || <big>f</big> || <big>c</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>c</big> || <big>b</big> || <big>e</big> || <big>a</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>c</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>c</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big> |} 

!style="background:#efefef;"| *+!style="background:#efefef;"| <big>e0</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f1</big>

!style="background:#efefef;"| <big>e0</big>| <big>e0</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f1</big>

!style="background:#efefef;"| <big>a1</big>| <big>a1</big> || <big>b</big> || <big>d</big> || <big>e</big> || <big>f</big> || <big>c</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>e</big> || <big>f</big> || <big>d</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>c</big> || <big>a</big> || <big>e</big> || <big>b</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e0</big>

!style="background:#efefef;"| *+!style="background:#efefef;"| <big>e0</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>e</big> || <big>d</big> || <big>a</big> || <big>c</big> || <big>b</big> |} {| border="1" cellpadding="4" align="center"!style="background:#efefef;"| *!style="background:#efefef;"| <big>e</big>!style="background:#efefef;"| <big>a</big>!style="background:#efefef;"| <big>b</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f2</big>

!style="background:#efefef;"| <big>e0</big>| <big>e0</big> || <big>a1</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f2</big>

!style="background:#efefef;"| <big>a1</big>| <big>a1</big> || <big>b2</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>e</big> || <big>f</big> || <big>a</big> || <big>d</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>e</big> || <big>b</big> || <big>c0</big>

!style="background:#efefef;"| <big>f2</big>| <big>f2</big> || <big>e0</big> || <big>d</big> || <big>b</big> || <big>c</big> || <big>a1</big>

!style="background:#efefef;"| *+!style="background:#efefef;"| <big>e0</big>!style="background:#efefef;"| <big>a1</big>!style="background:#efefef;"| <big>b2</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> || <big>e3</big>

!style="background:#efefef;"| <big>b0</big>| <big>b0</big> || <big>c1</big> || <big>a2</big> || <big>f</big> || <big>e</big> || <big>d3</big>

!style="background:#efefef;"| <big>c1</big>| <big>c1</big> || <big>d0</big> || <big>f3</big> || <big>e</big> || <big>a</big> || <big>b2</big>

!style="background:#efefef;"| <big>d2</big>| <big>d2</big> || <big>f3</big> || <big>e0</big> || <big>a</big> || <big>b</big> || <big>c1</big>

!style="background:#efefef;"| <big>f3</big>| <big>f3</big> || <big>e2</big> || <big>d1</big> || <big>b</big> || <big>c</big> || <big>a0</big>

!style="background:#efefef;"| *+!style="background:#efefef;"| <big>e(0,0)</big>!style="background:#efefef;"| <big>a(0,1)</big>!style="background:#efefef;"| <big>b(1,0)</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f(1,1)</big>

!style="background:#efefef;"| <big>e(0,0)</big>| <big>e(0,0)</big> || <big>a(0,1)</big> || <big>b(1,0)</big> || <big>c</big> || <big>d</big> || <big>f(1,1)</big>

!style="background:#efefef;"| <big>a(0,1)</big>| <big>a(0,1)</big> || <big>b(0,0)</big> || <big>c(1,1)</big> || <big>d</big> || <big>f</big> || <big>e(1,0)</big>

!style="background:#efefef;"| <big>b(1,0)</big>| <big>b(1,0)</big> || <big>c(1,1)</big> || <big>f(0,0)</big> || <big>e</big> || <big>a</big> || <big>d(0,1)</big>

!style="background:#efefef;"| <big>c(1,1)</big>| <big>c(1,1)</big> || <big>d(1,0)</big> || <big>e(0,1)</big> || <big>f</big> || <big>b</big> || <big>a</big> |-!style="background:#efefef;"| <big>d</big>| <big>d</big> || <big>f</big> || <big>a</big> || <big>b</big> || <big>e</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>e</big> || <big>d</big> || <big>a</big> || <big>c</big> || <big>b(0,0)</big>

!style="background:#efefef;"| *+!style="background:#efefef;"| <big>e0</big>!style="background:#efefef;"| <big>a1</big>!style="background:#efefef;"| <big>b2</big>!style="background:#efefef;"| <big>c3</big>!style="background:#efefef;"| <big>d</big>!style="background:#efefef;"| <big>f4</big>

!style="background:#efefef;"| <big>e0</big>| <big>e0</big> || <big>a1</big> || <big>b2</big> || <big>c3</big> || <big>d</big> || <big>f4</big>

!style="background:#efefef;"| <big>a1</big>| <big>a1</big> || <big>b2</big> || <big>c3</big> || <big>f4</big> || <big>e</big> || <big>d0</big>

!style="background:#efefef;"| <big>b2</big>| <big>b2</big> || <big>c3</big> || <big>e4</big> || <big>d0</big> || <big>f</big> || <big>a1</big>

!style="background:#efefef;"| <big>c3</big>| <big>c3</big> || <big>f4</big> || <big>d0</big> || <big>e1</big> || <big>a</big> || <big>b2</big>

!style="background:#efefef;"| <big>d4</big>| <big>d4</big> || <big>e0</big> || <big>f1</big> || <big>a2</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>d</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e3</big>

!style="background:#efefef;"| *+!style="background:#efefef;"| <big>e0</big>!style="background:#efefef;"| <big>a1</big>!style="background:#efefef;"| <big>b2</big>!style="background:#efefef;"| <big>c3</big>!style="background:#efefef;"| <big>d4</big>!style="background:#efefef;"| <big>f5</big>

!style="background:#efefef;"| <big>e0</big>| <big>e0</big> || <big>a1</big> || <big>b2</big> || <big>c3</big> || <big>d4</big> || <big>f5</big>

!style="background:#efefef;"| <big>a1</big>| <big>a1</big> || <big>b2</big> || <big>c3</big> || <big>f4</big> || <big>e5</big> || <big>d0</big>

!style="background:#efefef;"| <big>b2</big>| <big>b2</big> || <big>c3</big> || <big>e4</big> || <big>d5</big> || <big>f0</big> || <big>a1</big>

!style="background:#efefef;"| <big>c3</big>| <big>c3</big> || <big>f4</big> || <big>d5</big> || <big>a0</big> || <big>b1</big> || <big>e2</big>

!style="background:#efefef;"| <big>d4</big>| <big>d4</big> || <big>e5</big> || <big>f0</big> || <big>b1</big> || <big>a2</big> || <big>c3</big>

!style="background:#efefef;"| <big>f5</big>| <big>f5</big> || <big>d0</big> || <big>a1</big> || <big>e2</big> || <big>c3</big> || <big>b4</big>

| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>f</big> |-!style="background:#efefef;"| <big>a</big>| <big>aaa</big> || <big>b</big> || <big>c</big> || <big>f</big> || <big>e</big> || <big>d</big> |-!style="background:#efefef;"| <big>b</big>| <big>b</big> || <big>c</big> || <big>a</big> || <big>d</big> || <big>f</big> || <big>e</big> |-!style="background:#efefef;"| <big>c</big>| <big>c</big> || <big>f</big> || <big>d</big> || <big>e</big> || <big>a</big> || <big>b</big>

!style="background:#efefef;"| <big>ba</big>!style="background:#efefef;"| <big>c</big>!style="background:#efefef;"| <big>daa</big>!style="background:#efefef;"| <big>f</big>|-!style="background:#efefef;"| <big>e</big>| <big>e</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>d</big> || <big>fb</big>

!style="background:#efefef;"| <big>aaa</big>| <big>aaa</big> || <big>be</big> || <big>ca</big> || <big>fd</big> || <big>eb</big> || <big>dc</big>

| <big>b</big> || <big>cd</big> || <big>dc</big> || <big>e</big> || <big>faa</big> || <big>a</big>

| <big>c</big> || <big>fb</big> || <big>ed</big> || <big>da</big> || <big>ae</big> || <big>baa</big>

| <big>d</big> || <big>e</big> || <big>f</big> || <big>ac</big> || <big>b</big> || <big>c</big> |-!style="background:#efefef;"| <big>f</big>| <big>f</big> || <big>daa</big> || <big>a</big> || <big>b</big> || <big>c</big> || <big>e</big>