metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊1D12, C24⋊1D4, C42.18D6, C8⋊C4⋊3S3, C3⋊1(C8⋊3D4), (C2×C8).55D6, (C2×D24)⋊10C2, C4⋊D12⋊2C2, (C2×C12).36D4, C4.34(C2×D12), (C2×C4).25D12, C42⋊7S3⋊2C2, C12.277(C2×D4), C2.7(C8⋊D6), C6.7(C4⋊1D4), C6.4(C8⋊C22), (C4×C12).3C22, C2.9(C4⋊D12), (C2×C24).56C22, (C2×D12).6C22, C22.97(C2×D12), (C2×C12).733C23, (C2×Dic6).7C22, (C3×C8⋊C4)⋊4C2, (C2×C24⋊C2)⋊1C2, (C2×C6).116(C2×D4), (C2×C4).677(C22×S3), SmallGroup(192,271)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D12
G = < a,b,c | a8=b12=c2=1, bab-1=a5, cac=a3, cbc=b-1 >
Subgroups: 600 in 144 conjugacy classes, 47 normal (17 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, C12, D6, C2×C6, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C24⋊C2, D24, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C2×D12, C8⋊3D4, C3×C8⋊C4, C4⋊D12, C42⋊7S3, C2×C24⋊C2, C2×D24, C8⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4⋊1D4, C8⋊C22, C2×D12, C8⋊3D4, C4⋊D12, C8⋊D6, C8⋊D12
►On 96 points
(1 20 32 89 39 56 66 78)(2 57 33 79 40 21 67 90)(3 22 34 91 41 58 68 80)(4 59 35 81 42 23 69 92)(5 24 36 93 43 60 70 82)(6 49 25 83 44 13 71 94)(7 14 26 95 45 50 72 84)(8 51 27 73 46 15 61 96)(9 16 28 85 47 52 62 74)(10 53 29 75 48 17 63 86)(11 18 30 87 37 54 64 76)(12 55 31 77 38 19 65 88)
(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)(2 76)(3 75)(4 74)(5 73)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 50)(26 49)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)
G:=sub<Sym(96)| (1,20,32,89,39,56,66,78)(2,57,33,79,40,21,67,90)(3,22,34,91,41,58,68,80)(4,59,35,81,42,23,69,92)(5,24,36,93,43,60,70,82)(6,49,25,83,44,13,71,94)(7,14,26,95,45,50,72,84)(8,51,27,73,46,15,61,96)(9,16,28,85,47,52,62,74)(10,53,29,75,48,17,63,86)(11,18,30,87,37,54,64,76)(12,55,31,77,38,19,65,88), (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)(2,76)(3,75)(4,74)(5,73)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,50)(26,49)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)>;
G:=Group( (1,20,32,89,39,56,66,78)(2,57,33,79,40,21,67,90)(3,22,34,91,41,58,68,80)(4,59,35,81,42,23,69,92)(5,24,36,93,43,60,70,82)(6,49,25,83,44,13,71,94)(7,14,26,95,45,50,72,84)(8,51,27,73,46,15,61,96)(9,16,28,85,47,52,62,74)(10,53,29,75,48,17,63,86)(11,18,30,87,37,54,64,76)(12,55,31,77,38,19,65,88), (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)(2,76)(3,75)(4,74)(5,73)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,50)(26,49)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91) );
G=PermutationGroup([[(1,20,32,89,39,56,66,78),(2,57,33,79,40,21,67,90),(3,22,34,91,41,58,68,80),(4,59,35,81,42,23,69,92),(5,24,36,93,43,60,70,82),(6,49,25,83,44,13,71,94),(7,14,26,95,45,50,72,84),(8,51,27,73,46,15,61,96),(9,16,28,85,47,52,62,74),(10,53,29,75,48,17,63,86),(11,18,30,87,37,54,64,76),(12,55,31,77,38,19,65,88)], [(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),(2,76),(3,75),(4,74),(5,73),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,50),(26,49),(27,60),(28,59),(29,58),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C8⋊C22 | C8⋊D6 |
| kernel | C8⋊D12 | C3×C8⋊C4 | C4⋊D12 | C42⋊7S3 | C2×C24⋊C2 | C2×D24 | C8⋊C4 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 4 |
Matrix representation of C8⋊D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 60 | 46 | 19 |
| 0 | 0 | 13 | 70 | 54 | 27 |
| 0 | 0 | 17 | 0 | 16 | 13 |
| 0 | 0 | 0 | 17 | 60 | 3 |
| 72 | 70 | 0 | 0 | 0 | 0 |
| 25 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 46 | 16 | 3 |
| 0 | 0 | 27 | 54 | 70 | 13 |
| 0 | 0 | 3 | 16 | 46 | 27 |
| 0 | 0 | 57 | 60 | 46 | 19 |
| 72 | 70 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,13,17,0,0,0,60,70,0,17,0,0,46,54,16,60,0,0,19,27,13,3],[72,25,0,0,0,0,70,1,0,0,0,0,0,0,27,27,3,57,0,0,46,54,16,60,0,0,16,70,46,46,0,0,3,13,27,19],[72,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,72,0,0,0,0,0,72,1,0,0] >;
C8⋊D12 in GAP, Magma, Sage, TeX
C_8\rtimes D_{12} % in TeX
G:=Group("C8:D12"); // GroupNames label
G:=SmallGroup(192,271);
// by ID
G=gap.SmallGroup(192,271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations