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G = C6×C3⋊C8order 144 = 24·32

Direct product of C6 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C6×C3⋊C8, C6⋊C24, C12.3C12, C12.66D6, C62.4C4, C12.11Dic3, (C3×C6)⋊3C8, C32(C2×C24), C328(C2×C8), C4.14(S3×C6), (C2×C12).7C6, (C3×C12).8C4, (C2×C6).5C12, C6.5(C2×C12), C12.15(C2×C6), (C6×C12).10C2, (C2×C12).21S3, C4.3(C3×Dic3), C2.1(C6×Dic3), (C2×C6).9Dic3, C6.17(C2×Dic3), (C3×C12).44C22, C22.2(C3×Dic3), (C2×C4).5(C3×S3), (C3×C6).27(C2×C4), SmallGroup(144,74)

Series: Derived Chief Lower central Upper central

C1C3 — C6×C3⋊C8
C1C3C6C12C3×C12C3×C3⋊C8 — C6×C3⋊C8
C3 — C6×C3⋊C8
C1C2×C12

Generators and relations for C6×C3⋊C8
 G = < a,b,c | a6=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C6
2C6
3C8
3C8
2C12
2C12
2C2×C6
3C2×C8
2C2×C12
3C24
3C24
3C2×C24

Smallest permutation representation of C6×C3⋊C8
On 48 points
Generators in S48
(1 33 26 15 18 47)(2 34 27 16 19 48)(3 35 28 9 20 41)(4 36 29 10 21 42)(5 37 30 11 22 43)(6 38 31 12 23 44)(7 39 32 13 24 45)(8 40 25 14 17 46)
(1 26 18)(2 19 27)(3 28 20)(4 21 29)(5 30 22)(6 23 31)(7 32 24)(8 17 25)(9 41 35)(10 36 42)(11 43 37)(12 38 44)(13 45 39)(14 40 46)(15 47 33)(16 34 48)
(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)| (1,33,26,15,18,47)(2,34,27,16,19,48)(3,35,28,9,20,41)(4,36,29,10,21,42)(5,37,30,11,22,43)(6,38,31,12,23,44)(7,39,32,13,24,45)(8,40,25,14,17,46), (1,26,18)(2,19,27)(3,28,20)(4,21,29)(5,30,22)(6,23,31)(7,32,24)(8,17,25)(9,41,35)(10,36,42)(11,43,37)(12,38,44)(13,45,39)(14,40,46)(15,47,33)(16,34,48), (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( (1,33,26,15,18,47)(2,34,27,16,19,48)(3,35,28,9,20,41)(4,36,29,10,21,42)(5,37,30,11,22,43)(6,38,31,12,23,44)(7,39,32,13,24,45)(8,40,25,14,17,46), (1,26,18)(2,19,27)(3,28,20)(4,21,29)(5,30,22)(6,23,31)(7,32,24)(8,17,25)(9,41,35)(10,36,42)(11,43,37)(12,38,44)(13,45,39)(14,40,46)(15,47,33)(16,34,48), (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([(1,33,26,15,18,47),(2,34,27,16,19,48),(3,35,28,9,20,41),(4,36,29,10,21,42),(5,37,30,11,22,43),(6,38,31,12,23,44),(7,39,32,13,24,45),(8,40,25,14,17,46)], [(1,26,18),(2,19,27),(3,28,20),(4,21,29),(5,30,22),(6,23,31),(7,32,24),(8,17,25),(9,41,35),(10,36,42),(11,43,37),(12,38,44),(13,45,39),(14,40,46),(15,47,33),(16,34,48)], [(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)])

C6×C3⋊C8 is a maximal subgroup of
C6.(S3×C8)  C3⋊C8⋊Dic3  C2.Dic32  C12.77D12  C12.78D12  C6.16D24  C6.17D24  C6.Dic12  C12.73D12  C12.81D12  C12.15Dic6  C12.Dic6  C6.18D24  C12.82D12  Dic3×C24  D12.2Dic3  C3⋊C8.22D6  D12.27D6  S3×C2×C24

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A···6F6G···6O8A···8H12A···12H12I···12T24A···24P
order12223333344446···66···68···812···1212···1224···24
size11111122211111···12···23···31···12···23···3

72 irreducible representations

dim1111111111112222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C8C12C12C24S3Dic3D6Dic3C3×S3C3⋊C8C3×Dic3S3×C6C3×Dic3C3×C3⋊C8
kernelC6×C3⋊C8C3×C3⋊C8C6×C12C2×C3⋊C8C3×C12C62C3⋊C8C2×C12C3×C6C12C2×C6C6C2×C12C12C12C2×C6C2×C4C6C4C4C22C2
# reps12122242844161111242228

Matrix representation of C6×C3⋊C8 in GL4(𝔽73) generated by

65000
06500
00640
00064
,
1000
0100
0087
00064
,
63000
07200
00634
006610
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,7,64],[63,0,0,0,0,72,0,0,0,0,63,66,0,0,4,10] >;

C6×C3⋊C8 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes C_8
% in TeX

G:=Group("C6xC3:C8");
// GroupNames label

G:=SmallGroup(144,74);
// by ID

G=gap.SmallGroup(144,74);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,69,3461]);
// Polycyclic

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

Export

Subgroup lattice of C6×C3⋊C8 in TeX

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