Aliases: (C4×C12)⋊3C6, C4⋊D12⋊C3, C42⋊C3⋊3S3, C42⋊3(C3×S3), C3⋊(C23.A4), C22.1(S3×A4), (C22×S3).1A4, (C3×C42⋊C3)⋊3C2, (C2×C6).1(C2×A4), SmallGroup(288,405)
Series: Derived ►Chief ►Lower central ►Upper central
| C4×C12 — (C4×C12)⋊C6 |
Generators and relations for (C4×C12)⋊C6
G = < a,b,c | a4=b12=c6=1, ab=ba, cac-1=ab9, cbc-1=a-1b8 >
Subgroups: 498 in 59 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C12, A4, D6, C2×C6, C42, C2×D4, C3×S3, D12, C2×C12, C2×A4, C22×S3, C22×S3, C4⋊1D4, C3×A4, C42⋊C3, C42⋊C3, C4×C12, C2×D12, S3×A4, C23.A4, C4⋊D12, C3×C42⋊C3, (C4×C12)⋊C6
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, S3×A4, C23.A4, (C4×C12)⋊C6
Character table of (C4×C12)⋊C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 3 | 12 | 36 | 2 | 16 | 16 | 32 | 32 | 6 | 6 | 6 | 48 | 48 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 3 | 3 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ11 | 3 | 3 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ12 | 6 | -2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from C23.A4 |
| ρ13 | 6 | -2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from C23.A4 |
| ρ14 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | -2 | -3 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S3×A4 |
| ρ15 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 1-2√3 | -1 | -1 | 1+2√3 | orthogonal faithful |
| ρ16 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | -2 | 1 | 0 | 0 | -1 | 1+2√3 | 1-2√3 | -1 | orthogonal faithful |
| ρ17 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 1+2√3 | -1 | -1 | 1-2√3 | orthogonal faithful |
| ρ18 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | -2 | 1 | 0 | 0 | -1 | 1-2√3 | 1+2√3 | -1 | orthogonal faithful |
►On 36 points
(1 4 11 7)(2 5 12 8)(3 6 10 9)(13 16 19 22)(14 17 20 23)(15 18 21 24)
(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 22 28 8 15 29)(2 18 32 7 19 25)(3 14 36 9 23 33)(4 13 31 12 24 26)(5 21 35 11 16 34)(6 17 27 10 20 30)
G:=sub<Sym(36)| (1,4,11,7)(2,5,12,8)(3,6,10,9)(13,16,19,22)(14,17,20,23)(15,18,21,24), (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), (1,22,28,8,15,29)(2,18,32,7,19,25)(3,14,36,9,23,33)(4,13,31,12,24,26)(5,21,35,11,16,34)(6,17,27,10,20,30)>;
G:=Group( (1,4,11,7)(2,5,12,8)(3,6,10,9)(13,16,19,22)(14,17,20,23)(15,18,21,24), (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), (1,22,28,8,15,29)(2,18,32,7,19,25)(3,14,36,9,23,33)(4,13,31,12,24,26)(5,21,35,11,16,34)(6,17,27,10,20,30) );
G=PermutationGroup([[(1,4,11,7),(2,5,12,8),(3,6,10,9),(13,16,19,22),(14,17,20,23),(15,18,21,24)], [(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)], [(1,22,28,8,15,29),(2,18,32,7,19,25),(3,14,36,9,23,33),(4,13,31,12,24,26),(5,21,35,11,16,34),(6,17,27,10,20,30)]])
Matrix representation of (C4×C12)⋊C6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 11 | 2 | 9 | 1 | 0 | 0 |
| 7 | 6 | 9 | 4 | 0 | 0 |
| 4 | 0 | 10 | 2 | 10 | 7 |
| 1 | 1 | 0 | 2 | 6 | 3 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 4 | 9 | 1 | 11 | 0 | 0 |
| 9 | 9 | 8 | 11 | 0 | 0 |
| 3 | 12 | 7 | 9 | 7 | 10 |
| 8 | 12 | 2 | 9 | 3 | 10 |
| 3 | 8 | 3 | 0 | 6 | 12 |
| 4 | 9 | 0 | 0 | 6 | 7 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 11 | 0 | 6 | 0 | 0 | 10 |
| 2 | 0 | 7 | 7 | 5 | 10 |
| 5 | 0 | 11 | 3 | 5 | 9 |
G:=sub<GL(6,GF(13))| [1,0,11,7,4,1,0,1,2,6,0,1,0,0,9,9,10,0,0,0,1,4,2,2,0,0,0,0,10,6,0,0,0,0,7,3],[6,10,4,9,3,8,3,3,9,9,12,12,0,0,1,8,7,2,0,0,11,11,9,9,0,0,0,0,7,3,0,0,0,0,10,10],[3,4,10,11,2,5,8,9,7,0,0,0,3,0,0,6,7,11,0,0,0,0,7,3,6,6,0,0,5,5,12,7,0,10,10,9] >;
(C4×C12)⋊C6 in GAP, Magma, Sage, TeX
(C_4\times C_{12})\rtimes C_6 % in TeX
G:=Group("(C4xC12):C6"); // GroupNames label
G:=SmallGroup(288,405);
// by ID
G=gap.SmallGroup(288,405);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-3,-2,2,4664,198,772,4371,2110,360,1684,3036,5305]);
// Polycyclic
G:=Group<a,b,c|a^4=b^12=c^6=1,a*b=b*a,c*a*c^-1=a*b^9,c*b*c^-1=a^-1*b^8>;
// generators/relations
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