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The symmetric group on four letters, S4

The symmetric group on four letters, S4, contains the following permutations:

Permutations                                            Type

(12), (13), (14), (23), (24), (34)                         2-cycles

(12)(34), (13)(24), (14)(23)                     product of 2-cycles

(123), (124), (132), (134), (142), (143), (234), (243)     3-cycles

(1234), (1243), (1324), (1342), (1423), (1432)             4-cycles

Subgroups of S4

There are 30 subgroups of S4. Except for S4, their elements

are given in the following table:

Name             Elements                     Order

A4              {e, (12)(34), (13)(24), (14)(23), (123), (124),

(132), (134), (142), (143), (234), (243)}             12

V4         {e, (12)(34), (13)(24), (14)(23)}                    4

v1,v2,v3    {e, (12)(34)}, {e, (13)(24)}, {e, (14)(23)}         2,2,2

P1         {e, (123), (132)}                                     3

P2         {e, (124), (142)}                                     3

P3         {e, (134), (143)}                                     3

P4         {e, (234), (243)}                                     3

D {e, (12), (12)(34), (13)(24), (14)(23), (34), (1324), (1423)}   8

d    {e, (12)(34), (1324), (1423)}                              4

{e, (12)(34), (13), (13)(24), (14)(23), (24), (1234), (1432)}  8

{e, (13)(24), (1234), (1432)}                               4

{e, (12)(34), (13)(24), (14), (14)(23), (23), (1243), (1342)} 8

{e, (14)(23), (1243), (1342)}                               4

H1   {e, (12), (13), (23), (123), (132)}                         6

H2   {e, (12), (14), (24), (124), (142)}                         6

H3   {e, (13), (14), (34), (134), (143)}                         6

H4   {e, (23), (24), (34), (234), (243)}                         6

A   {e, (12), (12)(34), (34)}                                   4

a1,a2       {e, (12)} , { e, (34)}                                2,2

B         {e, (13), (13)(24), (24)}                             4

b1,b2       {e, (13)} , { e, (24)}                                2,2

C         {e, (14), (14)(23), (23)}                             4

c2,c1      {e, (14)} , { e, (23)}                                2,2