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