non-abelian, soluble, monomial
Aliases: C12⋊1S4, A4⋊1D12, C4⋊(C3⋊S4), C3⋊1(C4⋊S4), (C4×A4)⋊1S3, (C3×A4)⋊5D4, (C2×C6)⋊3D12, (C12×A4)⋊1C2, C6.30(C2×S4), (C22×C12)⋊2S3, (C2×A4).10D6, C22⋊(C12⋊S3), (C22×C6).21D6, (C6×A4).15C22, (C2×C3⋊S4)⋊3C2, C2.4(C2×C3⋊S4), C23.3(C2×C3⋊S3), (C22×C4)⋊2(C3⋊S3), SmallGroup(288,909)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊S4
G = < a,b,c,d,e | a12=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Subgroups: 1028 in 144 conjugacy classes, 27 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, A4, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, S4, C2×A4, C22×S3, C22×C6, C4⋊D4, C3×C12, C3×A4, C2×C3⋊S3, C4⋊Dic3, D6⋊C4, C4×A4, C2×D12, C2×C3⋊D4, C22×C12, C2×S4, C12⋊S3, C3⋊S4, C6×A4, C12⋊7D4, C4⋊S4, C12×A4, C2×C3⋊S4, C12⋊S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, D12, S4, C2×C3⋊S3, C2×S4, C12⋊S3, C3⋊S4, C4⋊S4, C2×C3⋊S4, C12⋊S4
Character table of C12⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 3 | 3 | 36 | 36 | 2 | 8 | 8 | 8 | 2 | 6 | 36 | 36 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 6 | 6 | 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 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 1 | 1 | √3 | -√3 | √3 | -√3 | 0 | -√3 | 0 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -2 | -√3 | √3 | -√3 | √3 | -√3 | -√3 | √3 | 0 | √3 | 0 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 1 | 1 | -√3 | √3 | -√3 | √3 | 0 | √3 | 0 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -2 | 1 | -√3 | √3 | -√3 | √3 | √3 | 0 | -√3 | √3 | 0 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -2 | 1 | √3 | -√3 | √3 | -√3 | -√3 | 0 | √3 | -√3 | 0 | √3 | orthogonal lifted from D12 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -2 | √3 | -√3 | √3 | -√3 | √3 | √3 | -√3 | 0 | -√3 | 0 | orthogonal lifted from D12 |
| ρ22 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 3 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ23 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ24 | 3 | 3 | -1 | -1 | -1 | 1 | 3 | 0 | 0 | 0 | -3 | 1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ25 | 3 | 3 | -1 | -1 | 1 | -1 | 3 | 0 | 0 | 0 | -3 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 6 | -2 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊S4 |
| ρ27 | 6 | -6 | -2 | 2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4⋊S4 |
| ρ28 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | -6 | 2 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3⋊S4 |
| ρ29 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | -3√3 | 3√3 | √3 | -√3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ30 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | 3√3 | -3√3 | -√3 | √3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►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 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 29 23)(2 30 24)(3 31 13)(4 32 14)(5 33 15)(6 34 16)(7 35 17)(8 36 18)(9 25 19)(10 26 20)(11 27 21)(12 28 22)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
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)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,29,23)(2,30,24)(3,31,13)(4,32,14)(5,33,15)(6,34,16)(7,35,17)(8,36,18)(9,25,19)(10,26,20)(11,27,21)(12,28,22), (2,12)(3,11)(4,10)(5,9)(6,8)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)>;
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)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,29,23)(2,30,24)(3,31,13)(4,32,14)(5,33,15)(6,34,16)(7,35,17)(8,36,18)(9,25,19)(10,26,20)(11,27,21)(12,28,22), (2,12)(3,11)(4,10)(5,9)(6,8)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28) );
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),(2,8),(3,9),(4,10),(5,11),(6,12),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,29,23),(2,30,24),(3,31,13),(4,32,14),(5,33,15),(6,34,16),(7,35,17),(8,36,18),(9,25,19),(10,26,20),(11,27,21),(12,28,22)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28)]])
Matrix representation of C12⋊S4 ►in GL5(𝔽13)
| 8 | 7 | 0 | 0 | 0 |
| 12 | 9 | 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 | 12 | 0 | 1 |
| 0 | 0 | 12 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 |
| 4 | 2 | 0 | 0 | 0 |
| 9 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [8,12,0,0,0,7,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[4,9,0,0,0,2,8,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
C12⋊S4 in GAP, Magma, Sage, TeX
C_{12}\rtimes S_4 % in TeX
G:=Group("C12:S4"); // GroupNames label
G:=SmallGroup(288,909);
// by ID
G=gap.SmallGroup(288,909);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,451,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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