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G = D6⋊C8order 96 = 25·3

The semidirect product of D6 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C8, C4.19D12, C12.52D4, C6.3M4(2), (C2×C8)⋊1S3, (C2×C24)⋊1C2, C2.5(S3×C8), C6.5(C2×C8), C31(C22⋊C8), (C2×C4).93D6, C2.1(D6⋊C4), C2.3(C8⋊S3), C4.27(C3⋊D4), C6.6(C22⋊C4), (C22×S3).2C4, C22.11(C4×S3), (C2×Dic3).4C4, (C2×C12).107C22, (C2×C3⋊C8)⋊9C2, (S3×C2×C4).7C2, (C2×C6).12(C2×C4), SmallGroup(96,27)

Series: Derived Chief Lower central Upper central

C1C6 — D6⋊C8
C1C3C6C12C2×C12S3×C2×C4 — D6⋊C8
C3C6 — D6⋊C8
C1C2×C4C2×C8

Generators and relations for D6⋊C8
 G = < a,b,c | a6=b2=c8=1, bab=a-1, ac=ca, cbc-1=a3b >

6C2
6C2
3C22
3C22
6C22
6C22
6C4
2S3
2S3
2C8
3C2×C4
3C23
6C8
6C2×C4
6C2×C4
2D6
2Dic3
2D6
3C2×C8
3C22×C4
2C3⋊C8
2C24
2C4×S3
2C4×S3
3C22⋊C8

Smallest permutation representation of D6⋊C8
On 48 points
Generators in S48
(1 15 43 31 22 39)(2 16 44 32 23 40)(3 9 45 25 24 33)(4 10 46 26 17 34)(5 11 47 27 18 35)(6 12 48 28 19 36)(7 13 41 29 20 37)(8 14 42 30 21 38)
(1 35)(2 48)(3 37)(4 42)(5 39)(6 44)(7 33)(8 46)(9 20)(10 14)(11 22)(12 16)(13 24)(15 18)(17 21)(19 23)(25 41)(26 38)(27 43)(28 40)(29 45)(30 34)(31 47)(32 36)
(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)

G:=sub<Sym(48)| (1,15,43,31,22,39)(2,16,44,32,23,40)(3,9,45,25,24,33)(4,10,46,26,17,34)(5,11,47,27,18,35)(6,12,48,28,19,36)(7,13,41,29,20,37)(8,14,42,30,21,38), (1,35)(2,48)(3,37)(4,42)(5,39)(6,44)(7,33)(8,46)(9,20)(10,14)(11,22)(12,16)(13,24)(15,18)(17,21)(19,23)(25,41)(26,38)(27,43)(28,40)(29,45)(30,34)(31,47)(32,36), (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)>;

G:=Group( (1,15,43,31,22,39)(2,16,44,32,23,40)(3,9,45,25,24,33)(4,10,46,26,17,34)(5,11,47,27,18,35)(6,12,48,28,19,36)(7,13,41,29,20,37)(8,14,42,30,21,38), (1,35)(2,48)(3,37)(4,42)(5,39)(6,44)(7,33)(8,46)(9,20)(10,14)(11,22)(12,16)(13,24)(15,18)(17,21)(19,23)(25,41)(26,38)(27,43)(28,40)(29,45)(30,34)(31,47)(32,36), (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) );

G=PermutationGroup([(1,15,43,31,22,39),(2,16,44,32,23,40),(3,9,45,25,24,33),(4,10,46,26,17,34),(5,11,47,27,18,35),(6,12,48,28,19,36),(7,13,41,29,20,37),(8,14,42,30,21,38)], [(1,35),(2,48),(3,37),(4,42),(5,39),(6,44),(7,33),(8,46),(9,20),(10,14),(11,22),(12,16),(13,24),(15,18),(17,21),(19,23),(25,41),(26,38),(27,43),(28,40),(29,45),(30,34),(31,47),(32,36)], [(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)])

D6⋊C8 is a maximal subgroup of
C42.282D6  C8×D12  C86D12  C42.243D6  C42.182D6  C89D12  C42.185D6  S3×C22⋊C8  C3⋊D4⋊C8  D6⋊M4(2)  D6⋊C8⋊C2  D62M4(2)  Dic3⋊M4(2)  C3⋊C826D4  D4⋊D12  D6.D8  D65SD16  D6.SD16  D6⋊C811C2  D43D12  D4.D12  C241C4⋊C2  D6.1SD16  Q83D12  Q8.11D12  D6⋊Q16  Q84D12  D6.Q16  D6⋊C8.C2  C8⋊Dic3⋊C2  C42.200D6  D12⋊C8  C42.202D6  D63M4(2)  C122M4(2)  C42.31D6  D6.2SD16  D6.4SD16  C4.Q8⋊S3  C6.(C4○D8)  D6.5D8  D6.2Q16  C2.D8⋊S3  C2.D87S3  C8×C3⋊D4  (C22×C8)⋊7S3  C2433D4  D66M4(2)  C24⋊D4  C2421D4  D6⋊C840C2  D12⋊D4  Dic6⋊D4  D66SD16  D68SD16  D127D4  Dic6.16D4  D65Q16  D12.17D4  D18⋊C8  C12.77D12  C12.78D12  C12.60D12  C60.94D4  D304C8  D303C8  Dic5.22D12  D30⋊C8
D6⋊C8 is a maximal quotient of
C4.8Dic12  C4.17D24  (C22×S3)⋊C8  (C2×Dic3)⋊C8  D122C8  Dic62C8  D6⋊C16  D12.C8  C8.25D12  Dic6.C8  (C2×C24)⋊5C4  D18⋊C8  C12.77D12  C12.78D12  C12.60D12  C60.94D4  D304C8  D303C8  Dic5.22D12  D30⋊C8

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D24A···24H
order1222223444444666888888881212121224···24
size11116621111662222222666622222···2

36 irreducible representations

dim1111111222222222
type++++++++
imageC1C2C2C2C4C4C8S3D4D6M4(2)D12C3⋊D4C4×S3S3×C8C8⋊S3
kernelD6⋊C8C2×C3⋊C8C2×C24S3×C2×C4C2×Dic3C22×S3D6C2×C8C12C2×C4C6C4C4C22C2C2
# reps1111228121222244

Matrix representation of D6⋊C8 in GL3(𝔽73) generated by

100
011
0720
,
100
07272
001
,
2200
0816
05765
G:=sub<GL(3,GF(73))| [1,0,0,0,1,72,0,1,0],[1,0,0,0,72,0,0,72,1],[22,0,0,0,8,57,0,16,65] >;

D6⋊C8 in GAP, Magma, Sage, TeX

D_6\rtimes C_8
% in TeX

G:=Group("D6:C8");
// GroupNames label

G:=SmallGroup(96,27);
// by ID

G=gap.SmallGroup(96,27);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,86,2309]);
// Polycyclic

G:=Group<a,b,c|a^6=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations

Export

Subgroup lattice of D6⋊C8 in TeX

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