metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.1C8, C8.31D12, C24.73D4, Dic6.1C8, (C2×C48)⋊2C2, (C2×C16)⋊2S3, C4.8(S3×C8), C3⋊1(D4.C8), C12.18(C2×C8), C2.9(D6⋊C8), C4○D12.1C4, C8○D12.3C2, (C2×C8).322D6, C12.C8⋊9C2, C4.41(D6⋊C4), C8.45(C3⋊D4), C6.8(C22⋊C8), (C2×C6).9M4(2), C4.Dic3.1C4, C12.56(C22⋊C4), (C2×C24).421C22, C22.2(C8⋊S3), (C2×C4).97(C4×S3), (C2×C12).231(C2×C4), SmallGroup(192,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.C8
G = < a,b,c | a12=b2=1, c8=a6, bab=a-1, ac=ca, cbc-1=a3b >
Smallest permutation representation of D12.C8
►On 96 points
Generators in S96
(1 70 50 90 26 47 9 78 58 82 18 39)(2 71 51 91 27 48 10 79 59 83 19 40)(3 72 52 92 28 33 11 80 60 84 20 41)(4 73 53 93 29 34 12 65 61 85 21 42)(5 74 54 94 30 35 13 66 62 86 22 43)(6 75 55 95 31 36 14 67 63 87 23 44)(7 76 56 96 32 37 15 68 64 88 24 45)(8 77 57 81 17 38 16 69 49 89 25 46)
(1 39)(2 59)(3 33)(4 53)(5 43)(6 63)(7 37)(8 57)(9 47)(10 51)(11 41)(12 61)(13 35)(14 55)(15 45)(16 49)(17 25)(18 70)(20 80)(21 29)(22 74)(24 68)(26 78)(28 72)(30 66)(32 76)(34 85)(36 95)(38 89)(40 83)(42 93)(44 87)(46 81)(48 91)(50 82)(52 92)(54 86)(56 96)(58 90)(60 84)(62 94)(64 88)(67 75)(71 79)
(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)
G:=sub<Sym(96)| (1,70,50,90,26,47,9,78,58,82,18,39)(2,71,51,91,27,48,10,79,59,83,19,40)(3,72,52,92,28,33,11,80,60,84,20,41)(4,73,53,93,29,34,12,65,61,85,21,42)(5,74,54,94,30,35,13,66,62,86,22,43)(6,75,55,95,31,36,14,67,63,87,23,44)(7,76,56,96,32,37,15,68,64,88,24,45)(8,77,57,81,17,38,16,69,49,89,25,46), (1,39)(2,59)(3,33)(4,53)(5,43)(6,63)(7,37)(8,57)(9,47)(10,51)(11,41)(12,61)(13,35)(14,55)(15,45)(16,49)(17,25)(18,70)(20,80)(21,29)(22,74)(24,68)(26,78)(28,72)(30,66)(32,76)(34,85)(36,95)(38,89)(40,83)(42,93)(44,87)(46,81)(48,91)(50,82)(52,92)(54,86)(56,96)(58,90)(60,84)(62,94)(64,88)(67,75)(71,79), (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)>;
G:=Group( (1,70,50,90,26,47,9,78,58,82,18,39)(2,71,51,91,27,48,10,79,59,83,19,40)(3,72,52,92,28,33,11,80,60,84,20,41)(4,73,53,93,29,34,12,65,61,85,21,42)(5,74,54,94,30,35,13,66,62,86,22,43)(6,75,55,95,31,36,14,67,63,87,23,44)(7,76,56,96,32,37,15,68,64,88,24,45)(8,77,57,81,17,38,16,69,49,89,25,46), (1,39)(2,59)(3,33)(4,53)(5,43)(6,63)(7,37)(8,57)(9,47)(10,51)(11,41)(12,61)(13,35)(14,55)(15,45)(16,49)(17,25)(18,70)(20,80)(21,29)(22,74)(24,68)(26,78)(28,72)(30,66)(32,76)(34,85)(36,95)(38,89)(40,83)(42,93)(44,87)(46,81)(48,91)(50,82)(52,92)(54,86)(56,96)(58,90)(60,84)(62,94)(64,88)(67,75)(71,79), (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) );
G=PermutationGroup([[(1,70,50,90,26,47,9,78,58,82,18,39),(2,71,51,91,27,48,10,79,59,83,19,40),(3,72,52,92,28,33,11,80,60,84,20,41),(4,73,53,93,29,34,12,65,61,85,21,42),(5,74,54,94,30,35,13,66,62,86,22,43),(6,75,55,95,31,36,14,67,63,87,23,44),(7,76,56,96,32,37,15,68,64,88,24,45),(8,77,57,81,17,38,16,69,49,89,25,46)], [(1,39),(2,59),(3,33),(4,53),(5,43),(6,63),(7,37),(8,57),(9,47),(10,51),(11,41),(12,61),(13,35),(14,55),(15,45),(16,49),(17,25),(18,70),(20,80),(21,29),(22,74),(24,68),(26,78),(28,72),(30,66),(32,76),(34,85),(36,95),(38,89),(40,83),(42,93),(44,87),(46,81),(48,91),(50,82),(52,92),(54,86),(56,96),(58,90),(60,84),(62,94),(64,88),(67,75),(71,79)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 12 | 2 | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | D4.C8 | D12.C8 |
| kernel | D12.C8 | C12.C8 | C2×C48 | C8○D12 | C4.Dic3 | C4○D12 | Dic6 | D12 | C2×C16 | C24 | C2×C8 | C2×C6 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of D12.C8 ►in GL2(𝔽97) generated by
| 68 | 29 |
| 68 | 39 |
| 68 | 29 |
| 58 | 29 |
| 3 | 64 |
| 33 | 36 |
G:=sub<GL(2,GF(97))| [68,68,29,39],[68,58,29,29],[3,33,64,36] >;
D12.C8 in GAP, Magma, Sage, TeX
D_{12}.C_8 % in TeX
G:=Group("D12.C8"); // GroupNames label
G:=SmallGroup(192,67);
// by ID
G=gap.SmallGroup(192,67);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,100,1123,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^2=1,c^8=a^6,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations
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