metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊3D8, C42.218D6, C3⋊C8⋊11D4, C4⋊1(D4⋊S3), C4⋊1D4⋊3S3, C3⋊2(C8⋊4D4), C4.14(S3×D4), C6.58(C2×D8), (C2×D4).56D6, C12.31(C2×D4), C4⋊D12⋊10C2, (C2×C12).148D4, C6.20(C4⋊1D4), (C6×D4).72C22, C2.11(C12⋊3D4), (C2×C12).391C23, (C4×C12).121C22, (C2×D12).105C22, (C4×C3⋊C8)⋊15C2, (C2×D4⋊S3)⋊14C2, (C3×C4⋊1D4)⋊2C2, C2.13(C2×D4⋊S3), (C2×C6).522(C2×D4), (C2×C3⋊C8).257C22, (C2×C4).131(C3⋊D4), (C2×C4).489(C22×S3), C22.195(C2×C3⋊D4), SmallGroup(192,632)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊D8
G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 592 in 162 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, D6, C2×C6, C2×C6, C42, C2×C8, D8, C2×D4, C2×D4, C3⋊C8, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×C8, C4⋊1D4, C4⋊1D4, C2×D8, C2×C3⋊C8, D4⋊S3, C4×C12, C2×D12, C2×D12, C6×D4, C6×D4, C8⋊4D4, C4×C3⋊C8, C4⋊D12, C2×D4⋊S3, C3×C4⋊1D4, C12⋊D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C2×D8, D4⋊S3, S3×D4, C2×C3⋊D4, C8⋊4D4, C2×D4⋊S3, C12⋊3D4, C12⋊D8
►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 77 95 58 47 28 23 70)(2 82 96 51 48 33 24 63)(3 75 85 56 37 26 13 68)(4 80 86 49 38 31 14 61)(5 73 87 54 39 36 15 66)(6 78 88 59 40 29 16 71)(7 83 89 52 41 34 17 64)(8 76 90 57 42 27 18 69)(9 81 91 50 43 32 19 62)(10 74 92 55 44 25 20 67)(11 79 93 60 45 30 21 72)(12 84 94 53 46 35 22 65)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 93)(14 92)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 96)(23 95)(24 94)(25 49)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 45)(38 44)(39 43)(40 42)(46 48)(61 74)(62 73)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)
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), (1,77,95,58,47,28,23,70)(2,82,96,51,48,33,24,63)(3,75,85,56,37,26,13,68)(4,80,86,49,38,31,14,61)(5,73,87,54,39,36,15,66)(6,78,88,59,40,29,16,71)(7,83,89,52,41,34,17,64)(8,76,90,57,42,27,18,69)(9,81,91,50,43,32,19,62)(10,74,92,55,44,25,20,67)(11,79,93,60,45,30,21,72)(12,84,94,53,46,35,22,65), (2,12)(3,11)(4,10)(5,9)(6,8)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,96)(23,95)(24,94)(25,49)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,45)(38,44)(39,43)(40,42)(46,48)(61,74)(62,73)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)>;
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), (1,77,95,58,47,28,23,70)(2,82,96,51,48,33,24,63)(3,75,85,56,37,26,13,68)(4,80,86,49,38,31,14,61)(5,73,87,54,39,36,15,66)(6,78,88,59,40,29,16,71)(7,83,89,52,41,34,17,64)(8,76,90,57,42,27,18,69)(9,81,91,50,43,32,19,62)(10,74,92,55,44,25,20,67)(11,79,93,60,45,30,21,72)(12,84,94,53,46,35,22,65), (2,12)(3,11)(4,10)(5,9)(6,8)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,96)(23,95)(24,94)(25,49)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,45)(38,44)(39,43)(40,42)(46,48)(61,74)(62,73)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75) );
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)], [(1,77,95,58,47,28,23,70),(2,82,96,51,48,33,24,63),(3,75,85,56,37,26,13,68),(4,80,86,49,38,31,14,61),(5,73,87,54,39,36,15,66),(6,78,88,59,40,29,16,71),(7,83,89,52,41,34,17,64),(8,76,90,57,42,27,18,69),(9,81,91,50,43,32,19,62),(10,74,92,55,44,25,20,67),(11,79,93,60,45,30,21,72),(12,84,94,53,46,35,22,65)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,93),(14,92),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,96),(23,95),(24,94),(25,49),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,45),(38,44),(39,43),(40,42),(46,48),(61,74),(62,73),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 24 | 24 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | C3⋊D4 | D4⋊S3 | S3×D4 |
| kernel | C12⋊D8 | C4×C3⋊C8 | C4⋊D12 | C2×D4⋊S3 | C3×C4⋊1D4 | C4⋊1D4 | C3⋊C8 | C2×C12 | C42 | C2×D4 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 4 | 2 |
Matrix representation of C12⋊D8 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 30 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 57 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 57 | 57 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,30,0,0,0,0,60,43,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,57,57,0,0,0,0,16,57],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C12⋊D8 in GAP, Magma, Sage, TeX
C_{12}\rtimes D_8 % in TeX
G:=Group("C12:D8"); // GroupNames label
G:=SmallGroup(192,632);
// by ID
G=gap.SmallGroup(192,632);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations