non-abelian, soluble, monomial
Aliases: (C2xC6):4S4, (C3xA4):8D4, C6.38(C2xS4), (C23xC6):4S3, A4:3(C3:D4), C6.7S4:4C2, C22:3(C3:S4), C24:2(C3:S3), (C22xA4):4S3, (C2xA4).12D6, C3:3(A4:D4), C22:(C32:7D4), (C22xC6).23D6, (C6xA4).17C22, (A4xC2xC6):3C2, (C2xC3:S4):4C2, C2.11(C2xC3:S4), (C2xC6):4(C3:D4), C23.5(C2xC3:S3), SmallGroup(288,917)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2xC6):4S4
G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, faf=ab3, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 952 in 156 conjugacy classes, 27 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, A4, D6, C2xC6, C2xC6, C22:C4, C2xD4, C24, C3:S3, C3xC6, C2xDic3, C3:D4, S4, C2xA4, C2xA4, C22xS3, C22xC6, C22xC6, C22wrC2, C3:Dic3, C3xA4, C2xC3:S3, C62, C6.D4, A4:C4, C2xC3:D4, C2xS4, C22xA4, C23xC6, C32:7D4, C3:S4, C6xA4, C6xA4, C24:4S3, A4:D4, C6.7S4, C2xC3:S4, A4xC2xC6, (C2xC6):4S4
Quotients: C1, C2, C22, S3, D4, D6, C3:S3, C3:D4, S4, C2xC3:S3, C2xS4, C32:7D4, C3:S4, A4:D4, C2xC3:S4, (C2xC6):4S4
Character table of (C2xC6):4S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 36 | 2 | 8 | 8 | 8 | 36 | 36 | 36 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -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 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | 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 |
| ρ5 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | -1 | 2 | -2 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -2 | 1 | 1 | -2 | 2 | -1 | 1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 2 | -2 | 1 | -2 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | -√-3 | 1 | √-3 | -1 | √-3 | -√-3 | 1 | 1 | √-3 | √-3 | -√-3 | 0 | -√-3 | 1 | -2 | 0 | complex lifted from C3:D4 |
| ρ15 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | √-3 | 1 | -√-3 | -1 | -√-3 | √-3 | 1 | 1 | √-3 | 0 | -√-3 | √-3 | 0 | -2 | 1 | -√-3 | complex lifted from C3:D4 |
| ρ16 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 1 | -√-3 | √-3 | √-3 | √-3 | -√-3 | 1 | 1 | -√-3 | complex lifted from C3:D4 |
| ρ17 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | -√-3 | 1 | √-3 | -1 | √-3 | -√-3 | 1 | 1 | -√-3 | 0 | √-3 | -√-3 | 0 | -2 | 1 | √-3 | complex lifted from C3:D4 |
| ρ18 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | √-3 | 1 | -√-3 | -1 | -√-3 | √-3 | 1 | -2 | 0 | √-3 | 0 | -√-3 | -√-3 | 1 | 1 | √-3 | complex lifted from C3:D4 |
| ρ19 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | -√-3 | 1 | √-3 | -1 | √-3 | -√-3 | 1 | -2 | 0 | -√-3 | 0 | √-3 | √-3 | 1 | 1 | -√-3 | complex lifted from C3:D4 |
| ρ20 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | √-3 | 1 | -√-3 | -1 | -√-3 | √-3 | 1 | 1 | -√-3 | -√-3 | √-3 | 0 | √-3 | 1 | -2 | 0 | complex lifted from C3:D4 |
| ρ21 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 1 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | 1 | 1 | √-3 | complex lifted from C3:D4 |
| ρ22 | 3 | 3 | 3 | -1 | -1 | -1 | 1 | 3 | 0 | 0 | 0 | -1 | 1 | -1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ23 | 3 | 3 | -3 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | 0 | 1 | -1 | -1 | -3 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ24 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 1 | -1 | 1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ25 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | 3 | 0 | 0 | 0 | -1 | 1 | 1 | -3 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ26 | 6 | 6 | 6 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | -3 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3:S4 |
| ρ27 | 6 | 6 | -6 | -2 | -2 | 2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -3 | 3 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xC3:S4 |
| ρ28 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -6 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4:D4 |
| ρ29 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3√-3 | 3 | -3√-3 | 1 | √-3 | -√-3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3√-3 | 3 | 3√-3 | 1 | -√-3 | √-3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(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 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 10)(8 11)(9 12)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 34)(32 35)(33 36)
(1 4)(2 5)(3 6)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)
(1 26 9)(2 27 10)(3 28 11)(4 29 12)(5 30 7)(6 25 8)(13 21 33)(14 22 34)(15 23 35)(16 24 36)(17 19 31)(18 20 32)
(2 6)(3 5)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 14)(15 18)(16 17)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)
G:=sub<Sym(36)| (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30), (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), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36), (1,4)(2,5)(3,6)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27), (1,26,9)(2,27,10)(3,28,11)(4,29,12)(5,30,7)(6,25,8)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,14)(15,18)(16,17)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)>;
G:=Group( (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30), (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), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,10)(8,11)(9,12)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36), (1,4)(2,5)(3,6)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27), (1,26,9)(2,27,10)(3,28,11)(4,29,12)(5,30,7)(6,25,8)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,14)(15,18)(16,17)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31) );
G=PermutationGroup([[(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,36),(8,31),(9,32),(10,33),(11,34),(12,35),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30)], [(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)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,10),(8,11),(9,12),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(31,34),(32,35),(33,36)], [(1,4),(2,5),(3,6),(7,36),(8,31),(9,32),(10,33),(11,34),(12,35),(13,16),(14,17),(15,18),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27)], [(1,26,9),(2,27,10),(3,28,11),(4,29,12),(5,30,7),(6,25,8),(13,21,33),(14,22,34),(15,23,35),(16,24,36),(17,19,31),(18,20,32)], [(2,6),(3,5),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,14),(15,18),(16,17),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31)]])
Matrix representation of (C2xC6):4S4 ►in GL5(F13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
(C2xC6):4S4 in GAP, Magma, Sage, TeX
(C_2\times C_6)\rtimes_4S_4
% in TeX
G:=Group("(C2xC6):4S4"); // GroupNames label
G:=SmallGroup(288,917);
// by ID
G=gap.SmallGroup(288,917);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,451,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a*b^3,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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