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G = D12.C8order 192 = 26·3

1st non-split extension by D12 of C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.1C8, C8.31D12, C24.73D4, Dic6.1C8, (C2×C48)⋊2C2, (C2×C16)⋊2S3, C4.8(S3×C8), C31(D4.C8), C12.18(C2×C8), C2.9(D6⋊C8), C4○D12.1C4, C8○D12.3C2, (C2×C8).322D6, C12.C89C2, 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

C1C12 — D12.C8
C1C3C6C12C24C2×C24C8○D12 — D12.C8
C3C6C12 — D12.C8
C1C8C2×C8C2×C16

Generators and relations for D12.C8
 G = < a,b,c | a12=b2=1, c8=a6, bab=a-1, ac=ca, cbc-1=a3b >

2C2
12C2
6C22
6C4
2C6
4S3
3D4
3Q8
6C8
6D4
6C2×C4
2Dic3
2D6
2C16
3M4(2)
3C4○D4
6C16
6M4(2)
6C2×C8
2C3⋊D4
2C4×S3
2C3⋊C8
3M5(2)
3C8○D4
2C48
2C8⋊S3
2S3×C8
2C3⋊C16
3D4.C8

Smallest permutation representation of D12.C8
On 96 points
Generators in S96
(1 31 62 33 76 85 9 23 54 41 68 93)(2 32 63 34 77 86 10 24 55 42 69 94)(3 17 64 35 78 87 11 25 56 43 70 95)(4 18 49 36 79 88 12 26 57 44 71 96)(5 19 50 37 80 89 13 27 58 45 72 81)(6 20 51 38 65 90 14 28 59 46 73 82)(7 21 52 39 66 91 15 29 60 47 74 83)(8 22 53 40 67 92 16 30 61 48 75 84)
(1 93)(2 55)(3 87)(4 49)(5 81)(6 59)(7 91)(8 53)(9 85)(10 63)(11 95)(12 57)(13 89)(14 51)(15 83)(16 61)(17 78)(19 72)(20 28)(21 66)(23 76)(24 32)(25 70)(27 80)(29 74)(31 68)(33 54)(34 86)(35 64)(36 96)(37 58)(38 90)(39 52)(40 84)(41 62)(42 94)(43 56)(44 88)(45 50)(46 82)(47 60)(48 92)(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)| (1,31,62,33,76,85,9,23,54,41,68,93)(2,32,63,34,77,86,10,24,55,42,69,94)(3,17,64,35,78,87,11,25,56,43,70,95)(4,18,49,36,79,88,12,26,57,44,71,96)(5,19,50,37,80,89,13,27,58,45,72,81)(6,20,51,38,65,90,14,28,59,46,73,82)(7,21,52,39,66,91,15,29,60,47,74,83)(8,22,53,40,67,92,16,30,61,48,75,84), (1,93)(2,55)(3,87)(4,49)(5,81)(6,59)(7,91)(8,53)(9,85)(10,63)(11,95)(12,57)(13,89)(14,51)(15,83)(16,61)(17,78)(19,72)(20,28)(21,66)(23,76)(24,32)(25,70)(27,80)(29,74)(31,68)(33,54)(34,86)(35,64)(36,96)(37,58)(38,90)(39,52)(40,84)(41,62)(42,94)(43,56)(44,88)(45,50)(46,82)(47,60)(48,92)(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( (1,31,62,33,76,85,9,23,54,41,68,93)(2,32,63,34,77,86,10,24,55,42,69,94)(3,17,64,35,78,87,11,25,56,43,70,95)(4,18,49,36,79,88,12,26,57,44,71,96)(5,19,50,37,80,89,13,27,58,45,72,81)(6,20,51,38,65,90,14,28,59,46,73,82)(7,21,52,39,66,91,15,29,60,47,74,83)(8,22,53,40,67,92,16,30,61,48,75,84), (1,93)(2,55)(3,87)(4,49)(5,81)(6,59)(7,91)(8,53)(9,85)(10,63)(11,95)(12,57)(13,89)(14,51)(15,83)(16,61)(17,78)(19,72)(20,28)(21,66)(23,76)(24,32)(25,70)(27,80)(29,74)(31,68)(33,54)(34,86)(35,64)(36,96)(37,58)(38,90)(39,52)(40,84)(41,62)(42,94)(43,56)(44,88)(45,50)(46,82)(47,60)(48,92)(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([(1,31,62,33,76,85,9,23,54,41,68,93),(2,32,63,34,77,86,10,24,55,42,69,94),(3,17,64,35,78,87,11,25,56,43,70,95),(4,18,49,36,79,88,12,26,57,44,71,96),(5,19,50,37,80,89,13,27,58,45,72,81),(6,20,51,38,65,90,14,28,59,46,73,82),(7,21,52,39,66,91,15,29,60,47,74,83),(8,22,53,40,67,92,16,30,61,48,75,84)], [(1,93),(2,55),(3,87),(4,49),(5,81),(6,59),(7,91),(8,53),(9,85),(10,63),(11,95),(12,57),(13,89),(14,51),(15,83),(16,61),(17,78),(19,72),(20,28),(21,66),(23,76),(24,32),(25,70),(27,80),(29,74),(31,68),(33,54),(34,86),(35,64),(36,96),(37,58),(38,90),(39,52),(40,84),(41,62),(42,94),(43,56),(44,88),(45,50),(46,82),(47,60),(48,92),(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 2A2B2C 3 4A4B4C4D6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D16A···16H16I16J16K16L24A···24H48A···48P
order122234444666888888881212121216···161616161624···2448···48
size11212211212222111122121222222···2121212122···22···2

60 irreducible representations

dim1111111122222222222
type++++++++
imageC1C2C2C2C4C4C8C8S3D4D6M4(2)D12C3⋊D4C4×S3S3×C8C8⋊S3D4.C8D12.C8
kernelD12.C8C12.C8C2×C48C8○D12C4.Dic3C4○D12Dic6D12C2×C16C24C2×C8C2×C6C8C8C2×C4C4C22C3C1
# reps11112244121222244816

Matrix representation of D12.C8 in GL2(𝔽97) generated by

6829
6839
,
6829
5829
,
364
3336
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

Export

Subgroup lattice of D12.C8 in TeX

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