direct product, metabelian, soluble, monomial, A-group
Aliases: S3xC22:A4, (C23xC6):6C6, C22:2(S3xA4), C24:9(C3xS3), (S3xC24):2C3, (C22xS3):2A4, (C2xC6):(C2xA4), C3:(C2xC22:A4), (C3xC22:A4):5C2, SmallGroup(288,1038)
Series: Derived ►Chief ►Lower central ►Upper central
| C23xC6 — S3xC22:A4 |
Generators and relations for S3xC22:A4
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >
Subgroups: 1658 in 303 conjugacy classes, 24 normal (9 characteristic)
C1, C2, C3, C3, C22, C22, S3, S3, C6, C23, C32, A4, D6, C2xC6, C2xC6, C24, C24, C3xS3, C2xA4, C22xS3, C22xS3, C22xC6, C25, C3xA4, C22:A4, C22:A4, S3xC23, C23xC6, S3xA4, C2xC22:A4, S3xC24, C3xC22:A4, S3xC22:A4
Quotients: C1, C2, C3, S3, C6, A4, C3xS3, C2xA4, C22:A4, S3xA4, C2xC22:A4, S3xC22:A4
Character table of S3xC22:A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | |
| size | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 2 | 16 | 16 | 32 | 32 | 6 | 6 | 6 | 6 | 6 | 48 | 48 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | linear of order 6 |
| ρ7 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | complex lifted from C3xS3 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | complex lifted from C3xS3 |
| ρ10 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ12 | 3 | 3 | -1 | -3 | -1 | -1 | -1 | 1 | 1 | 1 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ13 | 3 | -1 | -1 | -3 | -1 | 3 | -1 | -3 | 1 | 1 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ14 | 3 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 3 | -1 | -1 | -3 | 3 | -1 | -1 | 1 | 1 | 1 | 1 | -3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ16 | 3 | -1 | -1 | -3 | -1 | -1 | 3 | 1 | -3 | 1 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ17 | 3 | -1 | 3 | -3 | -1 | -1 | -1 | 1 | 1 | -3 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ18 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ19 | 3 | -1 | -1 | 3 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ20 | 6 | -2 | -2 | 0 | -2 | 6 | -2 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 1 | -3 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from S3xA4 |
| ρ21 | 6 | 6 | -2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from S3xA4 |
| ρ22 | 6 | -2 | -2 | 0 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 1 | 1 | -3 | 1 | 1 | 0 | 0 | orthogonal lifted from S3xA4 |
| ρ23 | 6 | -2 | -2 | 0 | -2 | -2 | 6 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -3 | 1 | 0 | 0 | orthogonal lifted from S3xA4 |
| ρ24 | 6 | -2 | 6 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -3 | 0 | 0 | orthogonal lifted from S3xA4 |
►On 36 points
Generators in S36
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 7)(2 9)(3 8)(4 10)(5 12)(6 11)(13 19)(14 21)(15 20)(16 22)(17 24)(18 23)(25 31)(26 33)(27 32)(28 34)(29 36)(30 35)
(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)
G:=sub<Sym(36)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,9)(3,8)(4,10)(5,12)(6,11)(13,19)(14,21)(15,20)(16,22)(17,24)(18,23)(25,31)(26,33)(27,32)(28,34)(29,36)(30,35), (13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,9)(3,8)(4,10)(5,12)(6,11)(13,19)(14,21)(15,20)(16,22)(17,24)(18,23)(25,31)(26,33)(27,32)(28,34)(29,36)(30,35), (13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,7),(2,9),(3,8),(4,10),(5,12),(6,11),(13,19),(14,21),(15,20),(16,22),(17,24),(18,23),(25,31),(26,33),(27,32),(28,34),(29,36),(30,35)], [(13,22),(14,23),(15,24),(16,19),(17,20),(18,21),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36)], [(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36)], [(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24)]])
Matrix representation of S3xC22:A4 ►in GL8(Z)
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;
S3xC22:A4 in GAP, Magma, Sage, TeX
S_3\times C_2^2\rtimes A_4
% in TeX
G:=Group("S3xC2^2:A4"); // GroupNames label
G:=SmallGroup(288,1038);
// by ID
G=gap.SmallGroup(288,1038);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-3,198,94,1271,516,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=f^2=g^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,g*c*g^-1=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,g*d*g^-1=c,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations
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