metabelian, supersoluble, monomial
Aliases: C12.70D12, C6.2(D6⋊C4), (C2×C12).72D6, (C3×C12).29D4, C4.Dic3⋊3S3, C62.28(C2×C4), C12.25(C3⋊D4), (C6×C12).21C22, C32⋊3(C4.D4), C4.12(C3⋊D12), C3⋊1(C12.46D4), C2.3(C6.D12), C22.2(C6.D6), (C2×C4).2S32, (C2×C6).10(C4×S3), (C22×C3⋊S3).1C4, (C2×C12⋊S3).7C2, (C3×C4.Dic3)⋊14C2, (C3×C6).26(C22⋊C4), SmallGroup(288,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.70D12
G = < a,b,c | a12=c2=1, b12=a6, bab-1=cac=a-1, cbc=a9b11 >
Subgroups: 706 in 119 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, M4(2), C2×D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C2×C12, C2×C12, C22×S3, C4.D4, C3×C12, C2×C3⋊S3, C62, C4.Dic3, C3×M4(2), C2×D12, C3×C3⋊C8, C12⋊S3, C6×C12, C22×C3⋊S3, C12.46D4, C3×C4.Dic3, C2×C12⋊S3, C12.70D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4.D4, S32, D6⋊C4, C6.D6, C3⋊D12, C12.46D4, C6.D12, C12.70D12
►On 24 points - transitive group 24T617
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
(1 15)(2 20)(3 13)(4 18)(5 11)(6 16)(7 9)(8 14)(10 12)(17 23)(19 21)(22 24)
G:=sub<Sym(24)| (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,11)(6,16)(7,9)(8,14)(10,12)(17,23)(19,21)(22,24)>;
G:=Group( (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,11)(6,16)(7,9)(8,14)(10,12)(17,23)(19,21)(22,24) );
G=PermutationGroup([[(1,11,21,7,17,3,13,23,9,19,5,15),(2,16,6,20,10,24,14,4,18,8,22,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15),(2,20),(3,13),(4,18),(5,11),(6,16),(7,9),(8,14),(10,12),(17,23),(19,21),(22,24)]])
G:=TransitiveGroup(24,617);
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | ··· | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 36 | 36 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | C4.D4 | S32 | C3⋊D12 | C6.D6 | C12.46D4 | C12.70D12 |
| kernel | C12.70D12 | C3×C4.Dic3 | C2×C12⋊S3 | C22×C3⋊S3 | C4.Dic3 | C3×C12 | C2×C12 | C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 1 | 4 | 4 |
Matrix representation of C12.70D12 ►in GL4(𝔽73) generated by
| 14 | 7 | 0 | 0 |
| 66 | 7 | 0 | 0 |
| 0 | 0 | 14 | 7 |
| 0 | 0 | 66 | 7 |
| 0 | 0 | 59 | 66 |
| 0 | 0 | 7 | 14 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,14,66,0,0,7,7],[0,0,72,0,0,0,72,1,59,7,0,0,66,14,0,0],[72,0,0,0,72,1,0,0,0,0,0,72,0,0,72,0] >;
C12.70D12 in GAP, Magma, Sage, TeX
C_{12}._{70}D_{12} % in TeX
G:=Group("C12.70D12"); // GroupNames label
G:=SmallGroup(288,207);
// by ID
G=gap.SmallGroup(288,207);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,219,100,675,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=c^2=1,b^12=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^9*b^11>;
// generators/relations