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G = C12⋊C8order 96 = 25·3

1st semidirect product of C12 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C121C8, C12.7Q8, C4.16D12, C12.32D4, C42.2S3, C4.7Dic6, C6.5M4(2), C4⋊(C3⋊C8), C31(C4⋊C8), C6.7(C2×C8), C6.1(C4⋊C4), (C2×C12).7C4, (C4×C12).4C2, (C2×C4).89D6, (C2×C4).3Dic3, C2.1(C4⋊Dic3), C2.2(C4.Dic3), C22.8(C2×Dic3), (C2×C12).103C22, C2.3(C2×C3⋊C8), (C2×C3⋊C8).8C2, (C2×C6).26(C2×C4), SmallGroup(96,11)

Series: Derived Chief Lower central Upper central

C1C6 — C12⋊C8
C1C3C6C12C2×C12C2×C3⋊C8 — C12⋊C8
C3C6 — C12⋊C8
C1C2×C4C42

Generators and relations for C12⋊C8
 G = < a,b | a12=b8=1, bab-1=a-1 >

2C4
6C8
6C8
2C12
3C2×C8
3C2×C8
2C3⋊C8
2C3⋊C8
3C4⋊C8

Smallest permutation representation of C12⋊C8
Regular action on 96 points
Generators in S96
(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 40 32 90 52 22 76 64)(2 39 33 89 53 21 77 63)(3 38 34 88 54 20 78 62)(4 37 35 87 55 19 79 61)(5 48 36 86 56 18 80 72)(6 47 25 85 57 17 81 71)(7 46 26 96 58 16 82 70)(8 45 27 95 59 15 83 69)(9 44 28 94 60 14 84 68)(10 43 29 93 49 13 73 67)(11 42 30 92 50 24 74 66)(12 41 31 91 51 23 75 65)

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), (1,40,32,90,52,22,76,64)(2,39,33,89,53,21,77,63)(3,38,34,88,54,20,78,62)(4,37,35,87,55,19,79,61)(5,48,36,86,56,18,80,72)(6,47,25,85,57,17,81,71)(7,46,26,96,58,16,82,70)(8,45,27,95,59,15,83,69)(9,44,28,94,60,14,84,68)(10,43,29,93,49,13,73,67)(11,42,30,92,50,24,74,66)(12,41,31,91,51,23,75,65)>;

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), (1,40,32,90,52,22,76,64)(2,39,33,89,53,21,77,63)(3,38,34,88,54,20,78,62)(4,37,35,87,55,19,79,61)(5,48,36,86,56,18,80,72)(6,47,25,85,57,17,81,71)(7,46,26,96,58,16,82,70)(8,45,27,95,59,15,83,69)(9,44,28,94,60,14,84,68)(10,43,29,93,49,13,73,67)(11,42,30,92,50,24,74,66)(12,41,31,91,51,23,75,65) );

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

C12⋊C8 is a maximal subgroup of
C4.8Dic12  C242C8  C241C8  C4.17D24  C12.53D8  C12.39SD16  C4.Dic12  C12.47D8  C4.D24  C12.2D8  C12.57D8  C12.26Q16  C12.9D8  C12.5Q16  C12.10D8  C8×Dic6  C2412Q8  C8×D12  C86D12  C24⋊Q8  C89D12  S3×C4⋊C8  C42.200D6  C42.202D6  C12⋊M4(2)  C42.30D6  C42.31D6  C127M4(2)  C42.285D6  C42.270D6  C42.43D6  C42.187D6  C12.50D8  C12.38SD16  D4.3Dic6  D4×C3⋊C8  C42.47D6  C123M4(2)  C127D8  D4.1D12  D4.2D12  Q84Dic6  Q85Dic6  Q8.5Dic6  Q8×C3⋊C8  C42.210D6  Q82D12  Q8.6D12  C127Q16  C42.61D6  D12.23D4  Dic6.4Q8  D12.4Q8  C122D8  Dic69D4  C125SD16  D125Q8  D126Q8  C12⋊Q16  Dic65Q8  Dic66Q8  C36⋊C8  C12.81D12  C12.57D12  C60.13Q8  C605C8  C60⋊C8  Dic5.13D12
C12⋊C8 is a maximal quotient of
C242C8  C241C8  C12⋊C16  C24.1C8  (C2×C12)⋊3C8  C36⋊C8  C12.81D12  C12.57D12  C60.13Q8  C605C8  C60⋊C8  Dic5.13D12

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H6A6B6C8A···8H12A···12L
order12223444444446668···812···12
size11112111122222226···62···2

36 irreducible representations

dim111112222222222
type+++++--+-+
imageC1C2C2C4C8S3D4Q8Dic3D6M4(2)C3⋊C8Dic6D12C4.Dic3
kernelC12⋊C8C2×C3⋊C8C4×C12C2×C12C12C42C12C12C2×C4C2×C4C6C4C4C4C2
# reps121481112124224

Matrix representation of C12⋊C8 in GL5(𝔽73)

720000
017200
01000
00001
000720
,
100000
0544800
0291900
0003665
0006537

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0],[10,0,0,0,0,0,54,29,0,0,0,48,19,0,0,0,0,0,36,65,0,0,0,65,37] >;

C12⋊C8 in GAP, Magma, Sage, TeX

C_{12}\rtimes C_8
% in TeX

G:=Group("C12:C8");
// GroupNames label

G:=SmallGroup(96,11);
// by ID

G=gap.SmallGroup(96,11);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,86,2309]);
// Polycyclic

G:=Group<a,b|a^12=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C12⋊C8 in TeX

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