metabelian, supersoluble, monomial
Aliases: C12.82D12, C62.4Q8, C3⋊C8.1Dic3, C12.91(C4×S3), (C2×C6).7Dic6, (C3×C12).114D4, (C2×C12).288D6, C6.4(C4⋊Dic3), C4.13(S3×Dic3), C3⋊1(C24.C4), C4.Dic3.1S3, C12.81(C3⋊D4), C6.7(Dic3⋊C4), (C6×C12).37C22, C32⋊5(C8.C4), C12.12(C2×Dic3), C4.31(C3⋊D12), C3⋊1(C12.53D4), C12.58D6.3C2, C2.5(Dic3⋊Dic3), C22.1(C32⋊2Q8), (C3×C3⋊C8).1C4, (C2×C4).60S32, (C2×C3⋊C8).5S3, (C6×C3⋊C8).11C2, (C3×C6).24(C4⋊C4), (C3×C12).35(C2×C4), (C3×C4.Dic3).3C2, SmallGroup(288,225)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.82D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a6b11 >
Subgroups: 178 in 70 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C2×C12, C2×C12, C8.C4, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×C3⋊C8, C3×C3⋊C8, C32⋊4C8, C6×C12, C24.C4, C12.53D4, C6×C3⋊C8, C3×C4.Dic3, C12.58D6, C12.82D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C8.C4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C32⋊2Q8, C24.C4, C12.53D4, Dic3⋊Dic3, C12.82D12
►On 48 points
(1 23 21 19 17 15 13 11 9 7 5 3)(2 16 6 20 10 24 14 4 18 8 22 12)(25 35 45 31 41 27 37 47 33 43 29 39)(26 28 30 32 34 36 38 40 42 44 46 48)
(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 37 38 39 40 41 42 43 44 45 46 47 48)
(1 39 7 33 13 27 19 45)(2 38 8 32 14 26 20 44)(3 37 9 31 15 25 21 43)(4 36 10 30 16 48 22 42)(5 35 11 29 17 47 23 41)(6 34 12 28 18 46 24 40)
G:=sub<Sym(48)| (1,23,21,19,17,15,13,11,9,7,5,3)(2,16,6,20,10,24,14,4,18,8,22,12)(25,35,45,31,41,27,37,47,33,43,29,39)(26,28,30,32,34,36,38,40,42,44,46,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,39,7,33,13,27,19,45)(2,38,8,32,14,26,20,44)(3,37,9,31,15,25,21,43)(4,36,10,30,16,48,22,42)(5,35,11,29,17,47,23,41)(6,34,12,28,18,46,24,40)>;
G:=Group( (1,23,21,19,17,15,13,11,9,7,5,3)(2,16,6,20,10,24,14,4,18,8,22,12)(25,35,45,31,41,27,37,47,33,43,29,39)(26,28,30,32,34,36,38,40,42,44,46,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,39,7,33,13,27,19,45)(2,38,8,32,14,26,20,44)(3,37,9,31,15,25,21,43)(4,36,10,30,16,48,22,42)(5,35,11,29,17,47,23,41)(6,34,12,28,18,46,24,40) );
G=PermutationGroup([[(1,23,21,19,17,15,13,11,9,7,5,3),(2,16,6,20,10,24,14,4,18,8,22,12),(25,35,45,31,41,27,37,47,33,43,29,39),(26,28,30,32,34,36,38,40,42,44,46,48)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,39,7,33,13,27,19,45),(2,38,8,32,14,26,20,44),(3,37,9,31,15,25,21,43),(4,36,10,30,16,48,22,42),(5,35,11,29,17,47,23,41),(6,34,12,28,18,46,24,40)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12F | 12G | ··· | 12K | 24A | ··· | 24H | 24I | 24J | 24K | 24L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | Q8 | Dic3 | D6 | C4×S3 | D12 | C3⋊D4 | Dic6 | C8.C4 | C24.C4 | S32 | S3×Dic3 | C3⋊D12 | C32⋊2Q8 | C12.53D4 | C12.82D12 |
| kernel | C12.82D12 | C6×C3⋊C8 | C3×C4.Dic3 | C12.58D6 | C3×C3⋊C8 | C2×C3⋊C8 | C4.Dic3 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C12 | C12 | C12 | C2×C6 | C32 | C3 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 4 |
Matrix representation of C12.82D12 ►in GL6(𝔽73)
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 27 | 46 |
G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[10,0,0,0,0,0,0,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,22,0,0,0,0,22,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,27,27,0,0,0,0,0,46] >;
C12.82D12 in GAP, Magma, Sage, TeX
C_{12}._{82}D_{12} % in TeX
G:=Group("C12.82D12"); // GroupNames label
G:=SmallGroup(288,225);
// by ID
G=gap.SmallGroup(288,225);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^11>;
// generators/relations