metabelian, supersoluble, monomial
Aliases: C24.3D6, D6.6D12, D12.18D6, Dic6.15D6, Dic3.8D12, C8.1S32, C3⋊C8.1D6, C6.6(S3×D4), C24⋊C2⋊2S3, C8⋊S3⋊2S3, (C4×S3).3D6, (S3×C6).3D4, C6.6(C2×D12), (S3×Dic6)⋊2C2, C2.11(S3×D12), C32⋊5Q16⋊2C2, C3⋊1(C8.D6), C3⋊1(D4.D6), (C3×C24).4C22, (C3×Dic3).3D4, C32⋊3Q16⋊3C2, D12⋊5S3.1C2, D12.S3⋊2C2, (S3×C12).5C22, (C3×C12).48C23, (C3×D12).4C22, C32⋊3(C8.C22), C12.122(C22×S3), (C3×Dic6).2C22, C32⋊4Q8.2C22, C4.45(C2×S32), (C3×C8⋊S3)⋊4C2, (C3×C24⋊C2)⋊6C2, (C3×C6).32(C2×D4), (C3×C3⋊C8).1C22, SmallGroup(288,448)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.3D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a11, cac-1=a-1, cbc-1=b5 >
Subgroups: 530 in 129 conjugacy classes, 40 normal (all 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, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C8⋊S3, C24⋊C2, C24⋊C2, Dic12, D4.S3, C3⋊Q16, C3×M4(2), C3×SD16, C2×Dic6, C4○D12, D4⋊2S3, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3, D6⋊S3, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C32⋊4Q8, C8.D6, D4.D6, D12.S3, C32⋊3Q16, C3×C8⋊S3, C3×C24⋊C2, C32⋊5Q16, S3×Dic6, D12⋊5S3, C24.3D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C8.C22, S32, C2×D12, S3×D4, C2×S32, C8.D6, D4.D6, S3×D12, C24.3D6
►On 96 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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 47 5 43 9 39 13 35 17 31 21 27)(2 34 6 30 10 26 14 46 18 42 22 38)(3 45 7 41 11 37 15 33 19 29 23 25)(4 32 8 28 12 48 16 44 20 40 24 36)(49 78 69 82 65 86 61 90 57 94 53 74)(50 89 70 93 66 73 62 77 58 81 54 85)(51 76 71 80 67 84 63 88 59 92 55 96)(52 87 72 91 68 95 64 75 60 79 56 83)
(1 86 13 74)(2 85 14 73)(3 84 15 96)(4 83 16 95)(5 82 17 94)(6 81 18 93)(7 80 19 92)(8 79 20 91)(9 78 21 90)(10 77 22 89)(11 76 23 88)(12 75 24 87)(25 51 37 63)(26 50 38 62)(27 49 39 61)(28 72 40 60)(29 71 41 59)(30 70 42 58)(31 69 43 57)(32 68 44 56)(33 67 45 55)(34 66 46 54)(35 65 47 53)(36 64 48 52)
G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,47,5,43,9,39,13,35,17,31,21,27)(2,34,6,30,10,26,14,46,18,42,22,38)(3,45,7,41,11,37,15,33,19,29,23,25)(4,32,8,28,12,48,16,44,20,40,24,36)(49,78,69,82,65,86,61,90,57,94,53,74)(50,89,70,93,66,73,62,77,58,81,54,85)(51,76,71,80,67,84,63,88,59,92,55,96)(52,87,72,91,68,95,64,75,60,79,56,83), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,47,5,43,9,39,13,35,17,31,21,27)(2,34,6,30,10,26,14,46,18,42,22,38)(3,45,7,41,11,37,15,33,19,29,23,25)(4,32,8,28,12,48,16,44,20,40,24,36)(49,78,69,82,65,86,61,90,57,94,53,74)(50,89,70,93,66,73,62,77,58,81,54,85)(51,76,71,80,67,84,63,88,59,92,55,96)(52,87,72,91,68,95,64,75,60,79,56,83), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52) );
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,47,5,43,9,39,13,35,17,31,21,27),(2,34,6,30,10,26,14,46,18,42,22,38),(3,45,7,41,11,37,15,33,19,29,23,25),(4,32,8,28,12,48,16,44,20,40,24,36),(49,78,69,82,65,86,61,90,57,94,53,74),(50,89,70,93,66,73,62,77,58,81,54,85),(51,76,71,80,67,84,63,88,59,92,55,96),(52,87,72,91,68,95,64,75,60,79,56,83)], [(1,86,13,74),(2,85,14,73),(3,84,15,96),(4,83,16,95),(5,82,17,94),(6,81,18,93),(7,80,19,92),(8,79,20,91),(9,78,21,90),(10,77,22,89),(11,76,23,88),(12,75,24,87),(25,51,37,63),(26,50,38,62),(27,49,39,61),(28,72,40,60),(29,71,41,59),(30,70,42,58),(31,69,43,57),(32,68,44,56),(33,67,45,55),(34,66,46,54),(35,65,47,53),(36,64,48,52)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | ··· | 24H | 24I | 24J |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 12 | 2 | 2 | 4 | 2 | 6 | 12 | 36 | 36 | 2 | 2 | 4 | 12 | 24 | 4 | 12 | 2 | 2 | 4 | 4 | 4 | 12 | 24 | 4 | ··· | 4 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | D12 | D12 | C8.C22 | S32 | S3×D4 | C2×S32 | C8.D6 | D4.D6 | S3×D12 | C24.3D6 |
| kernel | C24.3D6 | D12.S3 | C32⋊3Q16 | C3×C8⋊S3 | C3×C24⋊C2 | C32⋊5Q16 | S3×Dic6 | D12⋊5S3 | C8⋊S3 | C24⋊C2 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | Dic6 | C4×S3 | D12 | Dic3 | D6 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C24.3D6 ►in GL4(𝔽73) generated by
| 23 | 55 | 0 | 0 |
| 18 | 5 | 0 | 0 |
| 17 | 38 | 50 | 55 |
| 32 | 31 | 18 | 68 |
| 13 | 28 | 2 | 1 |
| 13 | 28 | 1 | 2 |
| 66 | 6 | 55 | 50 |
| 66 | 7 | 55 | 50 |
| 12 | 10 | 0 | 0 |
| 22 | 61 | 0 | 0 |
| 26 | 50 | 10 | 61 |
| 42 | 62 | 51 | 63 |
G:=sub<GL(4,GF(73))| [23,18,17,32,55,5,38,31,0,0,50,18,0,0,55,68],[13,13,66,66,28,28,6,7,2,1,55,55,1,2,50,50],[12,22,26,42,10,61,50,62,0,0,10,51,0,0,61,63] >;
C24.3D6 in GAP, Magma, Sage, TeX
C_{24}._3D_6 % in TeX
G:=Group("C24.3D6"); // GroupNames label
G:=SmallGroup(288,448);
// by ID
G=gap.SmallGroup(288,448);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,58,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations