metabelian, supersoluble, monomial
Aliases: C24⋊14D6, C8⋊9S32, C3⋊C8⋊18D6, C8⋊S3⋊5S3, D6.5(C4×S3), (C4×S3).28D6, C3⋊S3⋊1M4(2), C3⋊2(S3×M4(2)), (C3×C24)⋊21C22, Dic3.8(C4×S3), (S3×Dic3).1C4, C6.D6.3C4, C32⋊2(C2×M4(2)), C12.29D6⋊8C2, D6.Dic3⋊11C2, (S3×C12).2C22, C12.136(C22×S3), (C3×C12).137C23, C32⋊4C8⋊26C22, C2.5(C4×S32), C6.3(S3×C2×C4), (C2×S32).4C4, (C4×S32).2C2, C4.83(C2×S32), (C8×C3⋊S3)⋊13C2, (C3×C3⋊C8)⋊23C22, (S3×C6).1(C2×C4), (C3×C8⋊S3)⋊10C2, (C3×C6).3(C22×C4), (C4×C3⋊S3).85C22, C3⋊Dic3.30(C2×C4), (C3×Dic3).2(C2×C4), (C2×C3⋊S3).26(C2×C4), SmallGroup(288,439)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊D6
G = < a,b,c | a24=b6=c2=1, bab-1=a5, cac=a17, cbc=b-1 >
Subgroups: 498 in 147 conjugacy classes, 52 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, M4(2), C22×C4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C8, C8⋊S3, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C3×C3⋊C8, C32⋊4C8, C3×C24, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, S3×M4(2), C12.29D6, D6.Dic3, C3×C8⋊S3, C8×C3⋊S3, C4×S32, C24⋊D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4×S3, C22×S3, C2×M4(2), S32, S3×C2×C4, C2×S32, S3×M4(2), 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 17)(2 10)(4 20)(5 13)(7 23)(8 16)(11 19)(14 22)(25 33)(27 43)(28 36)(30 46)(31 39)(34 42)(37 45)(40 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), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48)>;
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), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48) );
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)], [(1,17),(2,10),(4,20),(5,13),(7,23),(8,16),(11,19),(14,22),(25,33),(27,43),(28,36),(30,46),(31,39),(34,42),(37,45),(40,48)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H | 24I | 24J | 24K | 24L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 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 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 6 | 6 | 9 | 9 | 2 | 2 | 4 | 12 | 12 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 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 | M4(2) | C4×S3 | C4×S3 | S32 | C2×S32 | S3×M4(2) | C4×S32 | C24⋊D6 |
| kernel | C24⋊D6 | C12.29D6 | D6.Dic3 | C3×C8⋊S3 | C8×C3⋊S3 | C4×S32 | S3×Dic3 | C6.D6 | C2×S32 | C8⋊S3 | C3⋊C8 | C24 | C4×S3 | C3⋊S3 | Dic3 | D6 | C8 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 4 | 2 | 4 |
Matrix representation of C24⋊D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 3 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 5 | 0 | 0 |
| 0 | 0 | 72 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,0,0,0,0,0,3,22,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,58,72,0,0,0,0,5,15,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C24⋊D6 in GAP, Magma, Sage, TeX
C_{24}\rtimes D_6 % in TeX
G:=Group("C24:D6"); // GroupNames label
G:=SmallGroup(288,439);
// by ID
G=gap.SmallGroup(288,439);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,219,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^5,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations