metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊5D12, C24⋊23D4, C12⋊7SD16, C42.260D6, (C4×C8)⋊12S3, (C4×C24)⋊17C2, C3⋊1(C8⋊5D4), C4⋊1(C24⋊C2), C12⋊2Q8⋊1C2, (C2×C4).79D12, (C2×C8).317D6, C4.30(C2×D12), C6.4(C2×SD16), (C2×C12).376D4, C12.273(C2×D4), C4⋊D12.1C2, C6.3(C4⋊1D4), C2.5(C4⋊D12), (C2×D12).1C22, C22.89(C2×D12), (C2×C24).389C22, (C2×C12).722C23, (C4×C12).306C22, (C2×Dic6).2C22, (C2×C24⋊C2)⋊7C2, C2.7(C2×C24⋊C2), (C2×C6).105(C2×D4), (C2×C4).665(C22×S3), SmallGroup(192,252)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊5D12
G = < a,b,c | a8=b12=c2=1, ab=ba, cac=a3, cbc=b-1 >
Subgroups: 536 in 142 conjugacy classes, 55 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C24⋊C2, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C8⋊5D4, C4×C24, C12⋊2Q8, C4⋊D12, C2×C24⋊C2, C8⋊5D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C4⋊1D4, C2×SD16, C24⋊C2, C2×D12, C8⋊5D4, C4⋊D12, C2×C24⋊C2, C8⋊5D12
►On 96 points
(1 61 78 60 20 96 25 37)(2 62 79 49 21 85 26 38)(3 63 80 50 22 86 27 39)(4 64 81 51 23 87 28 40)(5 65 82 52 24 88 29 41)(6 66 83 53 13 89 30 42)(7 67 84 54 14 90 31 43)(8 68 73 55 15 91 32 44)(9 69 74 56 16 92 33 45)(10 70 75 57 17 93 34 46)(11 71 76 58 18 94 35 47)(12 72 77 59 19 95 36 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)(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 77)(26 76)(27 75)(28 74)(29 73)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 72)
G:=sub<Sym(96)| (1,61,78,60,20,96,25,37)(2,62,79,49,21,85,26,38)(3,63,80,50,22,86,27,39)(4,64,81,51,23,87,28,40)(5,65,82,52,24,88,29,41)(6,66,83,53,13,89,30,42)(7,67,84,54,14,90,31,43)(8,68,73,55,15,91,32,44)(9,69,74,56,16,92,33,45)(10,70,75,57,17,93,34,46)(11,71,76,58,18,94,35,47)(12,72,77,59,19,95,36,48), (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)(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,77)(26,76)(27,75)(28,74)(29,73)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72)>;
G:=Group( (1,61,78,60,20,96,25,37)(2,62,79,49,21,85,26,38)(3,63,80,50,22,86,27,39)(4,64,81,51,23,87,28,40)(5,65,82,52,24,88,29,41)(6,66,83,53,13,89,30,42)(7,67,84,54,14,90,31,43)(8,68,73,55,15,91,32,44)(9,69,74,56,16,92,33,45)(10,70,75,57,17,93,34,46)(11,71,76,58,18,94,35,47)(12,72,77,59,19,95,36,48), (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)(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,77)(26,76)(27,75)(28,74)(29,73)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,72) );
G=PermutationGroup([[(1,61,78,60,20,96,25,37),(2,62,79,49,21,85,26,38),(3,63,80,50,22,86,27,39),(4,64,81,51,23,87,28,40),(5,65,82,52,24,88,29,41),(6,66,83,53,13,89,30,42),(7,67,84,54,14,90,31,43),(8,68,73,55,15,91,32,44),(9,69,74,56,16,92,33,45),(10,70,75,57,17,93,34,46),(11,71,76,58,18,94,35,47),(12,72,77,59,19,95,36,48)], [(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),(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,77),(26,76),(27,75),(28,74),(29,73),(30,84),(31,83),(32,82),(33,81),(34,80),(35,79),(36,78),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,96),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,72)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | ··· | 2 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | D12 | D12 | C24⋊C2 |
| kernel | C8⋊5D12 | C4×C24 | C12⋊2Q8 | C4⋊D12 | C2×C24⋊C2 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 16 |
Matrix representation of C8⋊5D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 61 | 0 | 0 |
| 0 | 0 | 6 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 2 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 72 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,6,0,0,0,0,61,12,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,72,0,0,0,0,2,1,0,0,0,0,0,0,1,72,0,0,0,0,2,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;
C8⋊5D12 in GAP, Magma, Sage, TeX
C_8\rtimes_5D_{12} % in TeX
G:=Group("C8:5D12"); // GroupNames label
G:=SmallGroup(192,252);
// by ID
G=gap.SmallGroup(192,252);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations