metabelian, supersoluble, monomial
Aliases: C12.14D12, (C3×C12).30D4, (C2×C12).73D6, C6.33(D6⋊C4), (C2×Dic3).Dic3, C62.29(C2×C4), (C2×Dic6).5S3, (C6×Dic3).1C4, (C6×Dic6).1C2, C4.Dic3.2S3, C12.26(C3⋊D4), C2.4(D6⋊Dic3), (C6×C12).22C22, C4.10(D6⋊S3), C4.20(C3⋊D12), C3⋊1(C12.10D4), C3⋊3(C12.47D4), C22.4(S3×Dic3), C12.58D6.2C2, C6.3(C6.D4), C32⋊3(C4.10D4), (C2×C4).3S32, (C2×C6).67(C4×S3), (C2×C6).5(C2×Dic3), (C3×C4.Dic3).1C2, (C3×C6).27(C22⋊C4), SmallGroup(288,208)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.14D12
G = < a,b,c | a12=1, b12=a6, c2=a3, bab-1=a-1, cac-1=a5, cbc-1=a3b11 >
Subgroups: 242 in 86 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4.10D4, C3×Dic3, C3×C12, C62, C4.Dic3, C4.Dic3, C3×M4(2), C2×Dic6, C6×Q8, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C6×Dic3, C6×C12, C12.47D4, C12.10D4, C3×C4.Dic3, C12.58D6, C6×Dic6, C12.14D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, C4.10D4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C12.47D4, C12.10D4, D6⋊Dic3, C12.14D12
►On 48 points
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)(25 47 45 43 41 39 37 35 33 31 29 27)(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 45 7 39 13 33 19 27)(2 38 20 44 14 26 8 32)(3 43 9 37 15 31 21 25)(4 36 22 42 16 48 10 30)(5 41 11 35 17 29 23 47)(6 34 24 40 18 46 12 28)
G:=sub<Sym(48)| (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12)(25,47,45,43,41,39,37,35,33,31,29,27)(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), (1,45,7,39,13,33,19,27)(2,38,20,44,14,26,8,32)(3,43,9,37,15,31,21,25)(4,36,22,42,16,48,10,30)(5,41,11,35,17,29,23,47)(6,34,24,40,18,46,12,28)>;
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)(25,47,45,43,41,39,37,35,33,31,29,27)(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), (1,45,7,39,13,33,19,27)(2,38,20,44,14,26,8,32)(3,43,9,37,15,31,21,25)(4,36,22,42,16,48,10,30)(5,41,11,35,17,29,23,47)(6,34,24,40,18,46,12,28) );
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),(25,47,45,43,41,39,37,35,33,31,29,27),(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)], [(1,45,7,39,13,33,19,27),(2,38,20,44,14,26,8,32),(3,43,9,37,15,31,21,25),(4,36,22,42,16,48,10,30),(5,41,11,35,17,29,23,47),(6,34,24,40,18,46,12,28)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12I | 12J | 12K | 12L | 12M | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 36 | 36 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | - | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | Dic3 | D6 | D12 | C3⋊D4 | C4×S3 | C4.10D4 | S32 | D6⋊S3 | C3⋊D12 | S3×Dic3 | C12.47D4 | C12.10D4 | C12.14D12 |
| kernel | C12.14D12 | C3×C4.Dic3 | C12.58D6 | C6×Dic6 | C6×Dic3 | C4.Dic3 | C2×Dic6 | C3×C12 | C2×Dic3 | C2×C12 | C12 | C12 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C12.14D12 ►in GL4(𝔽73) generated by
| 3 | 21 | 20 | 8 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 70 | 0 |
| 0 | 0 | 0 | 24 |
| 8 | 12 | 21 | 20 |
| 21 | 65 | 70 | 65 |
| 0 | 0 | 0 | 64 |
| 0 | 0 | 49 | 0 |
| 65 | 61 | 65 | 40 |
| 0 | 0 | 49 | 0 |
| 0 | 65 | 0 | 0 |
| 52 | 8 | 3 | 8 |
G:=sub<GL(4,GF(73))| [3,0,0,0,21,49,0,0,20,0,70,0,8,0,0,24],[8,21,0,0,12,65,0,0,21,70,0,49,20,65,64,0],[65,0,0,52,61,0,65,8,65,49,0,3,40,0,0,8] >;
C12.14D12 in GAP, Magma, Sage, TeX
C_{12}._{14}D_{12} % in TeX
G:=Group("C12.14D12"); // GroupNames label
G:=SmallGroup(288,208);
// by ID
G=gap.SmallGroup(288,208);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations