## Some group multiplication tables

The Klein four group

```Described via                   2        2
generators a,b with relations  a  = e,  b  = e,  ba = ab

|
|   e      a      b      ab
_____|____________________________
|
e  |   e      a      b      ab
|
a  |   a      e     ab       b
|
b  |   b     ab      e       a
|
ab  |   ab     b      a       e

```

The cyclic group of order 6

```Described via the               6
generator  a   with relation   a  = e

|                  2      3      4      5
|   e      a      a      a      a      a
_____|_________________________________________
|
|                  2      3      4      5
e  |   e      a      a      a      a      a
|
|           2      3      4      5
a  |   a      a      a      a      a      e
|
2 |    2      3      4      5
a  |   a      a      a      a      e      a
|
3 |    3      4      5                    2
a  |   a      a      a      e      a      a
|
4 |    4      5                    2      3
a  |   a      a      e      a      a      a
|
5 |    5                    2      3      4
a  |   a      e      a      a      a      a

```

The symmetric group on three elements

```Described via                   3        2             -1
generators a,b with relations  a  = e,  b  = e,  ba = a  b

|                  2                   2
|   e      a      a      b      ab    a b
_____|__________________________________________
|
|                  2                   2
e  |   e      a      a      b      ab    a b
|
|           2                   2
a  |   a      a      e      ab    a b     b
|
2 |    2                    2
a  |   a      e      a      a b    b      ab
|
|          2                     2
b  |   b     a b     ab      e     a      a
|
|                 2                     2
ab  |   ab     b     a b      a     e      a
|
2  |   2                      2
a b |  a b     ab     b       a      a     e

```

The dihedral group of order eight

```Described via                   4        2             -1
generators a,b with relations  a  = e,  b  = e,  ba = a  b

|                  2      3                   2      3
|   e      a      a      a      b      ab    a b    a b
_____|________________________________________________________
|
|                  2      3                   2      3
e  |   e      a      a      a      b      ab    a b    a b
|
|           2      3                   2      3
a  |   a      a      a      e      ab    a b    a b     b
|
2 |    2      3                   2      3
a  |   a      a      e      a     a b    a b     b      ab
|
3 |    3                    2     3                    2
a  |   a      e      a      a     a b     b      ab    a b
|
|          3      2                     3      2
b  |   b     a b    a b     ab     e      a      a      a
|
|                 3      2                     3      2
ab  |   ab     b     a b    a b     a      e      a      a
|
2  |   2                    3       2                    3
a b |  a b     ab     b     a b     a      a      e      a
|
3  |   3      2                     3      2
a b |  a b    a b     ab     b      a      a      a      e

```

The quaternion group (of order eight)

```Described via                   4        2    2         -1
generators a,b with relations  a  = e,  b  = a ,  ba = a  b

|                  2      3                   2      3
|   e      a      a      a      b      ab    a b    a b
_____|________________________________________________________
|
|                  2      3                   2      3
e  |   e      a      a      a      b      ab    a b    a b
|
|           2      3                   2      3
a  |   a      a      a      e      ab    a b    a b     b
|
2 |    2      3                   2      3
a  |   a      a      e      a     a b    a b     b      ab
|
3 |    3                    2     3                    2
a  |   a      e      a      a     a b     b      ab    a b
|
|          3      2              2                    3
b  |   b     a b    a b     ab     a      a      e      a
|
|                 3      2       3      2
ab  |   ab     b     a b    a b     a      a      a      e
|
2  |   2                    3              3      2
a b |  a b     ab     b     a b     e      a      a      a
|
3  |   3      2                                   3      2
a b |  a b    a b     ab     b      a      e      a      a

```