metabelian, supersoluble, monomial
Aliases: C24.46D6, C8.19S32, C3⋊C8.20D6, C8⋊S3⋊6S3, D6.1(C4×S3), (C4×S3).30D6, C32⋊3(C8○D4), C3⋊2(D12.C4), C3⋊D12.2C4, D6⋊S3.2C4, Dic3.2(C4×S3), C32⋊2Q8.2C4, (S3×C12).7C22, (C3×C24).47C22, D6.D6.3C2, C12.31D6⋊10C2, C12.139(C22×S3), (C3×C12).140C23, C32⋊4C8.37C22, C2.8(C4×S32), C6.6(S3×C2×C4), (S3×C3⋊C8)⋊11C2, C4.86(C2×S32), (C8×C3⋊S3)⋊14C2, (S3×C6).3(C2×C4), (C3×C8⋊S3)⋊12C2, (C3×C3⋊C8).25C22, (C3×C6).6(C22×C4), (C4×C3⋊S3).87C22, C3⋊Dic3.32(C2×C4), (C3×Dic3).3(C2×C4), (C2×C3⋊S3).28(C2×C4), SmallGroup(288,453)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=a5, cac-1=a17, cbc-1=a12b-1 >
Subgroups: 434 in 135 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, M4(2), C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C8⋊S3, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C3×C3⋊C8, C32⋊4C8, C3×C24, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, D12.C4, S3×C3⋊C8, C12.31D6, C3×C8⋊S3, C8×C3⋊S3, D6.D6, C24.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S32, S3×C2×C4, C2×S32, D12.C4, C4×S32, C24.D6
►On 48 points
(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 26 17 34 9 42)(2 31 18 39 10 47)(3 36 19 44 11 28)(4 41 20 25 12 33)(5 46 21 30 13 38)(6 27 22 35 14 43)(7 32 23 40 15 48)(8 37 24 45 16 29)
(1 23 13 11)(2 16 14 4)(3 9 15 21)(5 19 17 7)(6 12 18 24)(8 22 20 10)(25 27 37 39)(26 44 38 32)(28 30 40 42)(29 47 41 35)(31 33 43 45)(34 36 46 48)
G:=sub<Sym(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), (1,26,17,34,9,42)(2,31,18,39,10,47)(3,36,19,44,11,28)(4,41,20,25,12,33)(5,46,21,30,13,38)(6,27,22,35,14,43)(7,32,23,40,15,48)(8,37,24,45,16,29), (1,23,13,11)(2,16,14,4)(3,9,15,21)(5,19,17,7)(6,12,18,24)(8,22,20,10)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,26,17,34,9,42)(2,31,18,39,10,47)(3,36,19,44,11,28)(4,41,20,25,12,33)(5,46,21,30,13,38)(6,27,22,35,14,43)(7,32,23,40,15,48)(8,37,24,45,16,29), (1,23,13,11)(2,16,14,4)(3,9,15,21)(5,19,17,7)(6,12,18,24)(8,22,20,10)(25,27,37,39)(26,44,38,32)(28,30,40,42)(29,47,41,35)(31,33,43,45)(34,36,46,48) );
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),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,26,17,34,9,42),(2,31,18,39,10,47),(3,36,19,44,11,28),(4,41,20,25,12,33),(5,46,21,30,13,38),(6,27,22,35,14,43),(7,32,23,40,15,48),(8,37,24,45,16,29)], [(1,23,13,11),(2,16,14,4),(3,9,15,21),(5,19,17,7),(6,12,18,24),(8,22,20,10),(25,27,37,39),(26,44,38,32),(28,30,40,42),(29,47,41,35),(31,33,43,45),(34,36,46,48)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H | 24I | 24J | 24K | 24L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 6 | 18 | 2 | 2 | 4 | 1 | 1 | 6 | 6 | 18 | 2 | 2 | 4 | 12 | 12 | 2 | 2 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | S32 | C2×S32 | D12.C4 | C4×S32 | C24.D6 |
| kernel | C24.D6 | S3×C3⋊C8 | C12.31D6 | C3×C8⋊S3 | C8×C3⋊S3 | D6.D6 | D6⋊S3 | C3⋊D12 | C32⋊2Q8 | C8⋊S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | C32 | C8 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 4 | 2 | 4 |
Matrix representation of C24.D6 ►in GL6(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 63 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 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 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,63,0,0,0,0,0,0,10,0,0,0,0,0,0,46,46,0,0,0,0,27,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;
C24.D6 in GAP, Magma, Sage, TeX
C_{24}.D_6 % in TeX
G:=Group("C24.D6"); // GroupNames label
G:=SmallGroup(288,453);
// by ID
G=gap.SmallGroup(288,453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,219,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^5,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations