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G = D6⋊C8  order 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), C3⋊1(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

Derived series
C1 — C6 — D6⋊C8
Chief series
C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — D6⋊C8
Lower central
C3 — C6 — D6⋊C8
Upper central
C1 — C2×C4 — C2×C8

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

C1
C2
C2
C2
6C2
6C2
C3
C22
C4
C4
3C22
3C22
6C22
6C22
6C4
C6
C6
C6
2S3
2S3
C2×C4
2C8
3C2×C4
3C23
6C8
6C2×C4
6C2×C4
C2×C6
C12
C12
D6
D6
2D6
2Dic3
2D6
C2×C8
3C2×C8
3C22×C4
C22×S3
C2×Dic3
C2×C12
2C3⋊C8
2C24
2C4×S3
2C4×S3
3C22⋊C8
C2×C24
S3×C2×C4
C2×C3⋊C8
D6⋊C8

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

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

(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)| (1,32,43,11,19,39)(2,25,44,12,20,40)(3,26,45,13,21,33)(4,27,46,14,22,34)(5,28,47,15,23,35)(6,29,48,16,24,36)(7,30,41,9,17,37)(8,31,42,10,18,38), (1,35)(2,48)(3,37)(4,42)(5,39)(6,44)(7,33)(8,46)(9,45)(10,34)(11,47)(12,36)(13,41)(14,38)(15,43)(16,40)(17,26)(18,22)(19,28)(20,24)(21,30)(23,32)(25,29)(27,31), (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( (1,32,43,11,19,39)(2,25,44,12,20,40)(3,26,45,13,21,33)(4,27,46,14,22,34)(5,28,47,15,23,35)(6,29,48,16,24,36)(7,30,41,9,17,37)(8,31,42,10,18,38), (1,35)(2,48)(3,37)(4,42)(5,39)(6,44)(7,33)(8,46)(9,45)(10,34)(11,47)(12,36)(13,41)(14,38)(15,43)(16,40)(17,26)(18,22)(19,28)(20,24)(21,30)(23,32)(25,29)(27,31), (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([[(1,32,43,11,19,39),(2,25,44,12,20,40),(3,26,45,13,21,33),(4,27,46,14,22,34),(5,28,47,15,23,35),(6,29,48,16,24,36),(7,30,41,9,17,37),(8,31,42,10,18,38)], [(1,35),(2,48),(3,37),(4,42),(5,39),(6,44),(7,33),(8,46),(9,45),(10,34),(11,47),(12,36),(13,41),(14,38),(15,43),(16,40),(17,26),(18,22),(19,28),(20,24),(21,30),(23,32),(25,29),(27,31)], [(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  C8⋊6D12  C42.243D6  C42.182D6  C8⋊9D12  C42.185D6  S3×C22⋊C8  C3⋊D4⋊C8  D6⋊M4(2)  D6⋊C8⋊C2  D6⋊2M4(2)  Dic3⋊M4(2)  C3⋊C8⋊26D4  D4⋊D12  D6.D8  D6⋊5SD16  D6.SD16  D6⋊C8⋊11C2  D4⋊3D12  D4.D12  C24⋊1C4⋊C2  D6.1SD16  Q8⋊3D12  Q8.11D12  D6⋊Q16  Q8⋊4D12  D6.Q16  D6⋊C8.C2  C8⋊Dic3⋊C2  C42.200D6  D12⋊C8  C42.202D6  D6⋊3M4(2)  C12⋊2M4(2)  C42.31D6  D6.2SD16  D6.4SD16  C4.Q8⋊S3  C6.(C4○D8)  D6.5D8  D6.2Q16  C2.D8⋊S3  C2.D8⋊7S3  C8×C3⋊D4  (C22×C8)⋊7S3  C24⋊33D4  D6⋊6M4(2)  C24⋊D4  C24⋊21D4  D6⋊C8⋊40C2  D12⋊D4  Dic6⋊D4  D6⋊6SD16  D6⋊8SD16  D12⋊7D4  Dic6.16D4  D6⋊5Q16  D12.17D4  D18⋊C8  C12.77D12  C12.78D12  C12.60D12  C60.94D4  D30⋊4C8  D30⋊3C8  Dic5.22D12  D30⋊C8
D6⋊C8 is a maximal quotient of
 C4.8Dic12  C4.17D24  (C22×S3)⋊C8  (C2×Dic3)⋊C8  D12⋊2C8  Dic6⋊2C8  D6⋊C16  D12.C8  C8.25D12  Dic6.C8  (C2×C24)⋊5C4  D18⋊C8  C12.77D12  C12.78D12  C12.60D12  C60.94D4  D30⋊4C8  D30⋊3C8  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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