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G = C6×C3⋊D4  order 144 = 24·32

Direct product of C6 and C3⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C6×C3⋊D4, C62⋊7C22, (C2×C6)⋊9D6, C3⋊3(C6×D4), (C3×C6)⋊6D4, C6⋊2(C3×D4), D6⋊3(C2×C6), C22⋊4(S3×C6), (C2×C62)⋊2C2, C23⋊3(C3×S3), (C22×C6)⋊4C6, (C22×C6)⋊3S3, C32⋊12(C2×D4), (C22×S3)⋊4C6, (S3×C6)⋊9C22, (C2×Dic3)⋊4C6, Dic3⋊2(C2×C6), (C6×Dic3)⋊10C2, C6.10(C22×C6), (C3×C6).28C23, C6.49(C22×S3), (C3×Dic3)⋊9C22, (S3×C2×C6)⋊6C2, (C2×C6)⋊5(C2×C6), C2.10(S3×C2×C6), SmallGroup(144,167)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C6×C3⋊D4
Chief series
C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — C6×C3⋊D4
Lower central
C3 — C6 — C6×C3⋊D4
Upper central
C1 — C2×C6 — C22×C6

Generators and relations for C6×C3⋊D4
 G = < a,b,c,d | a6=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 248 in 124 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×Dic3, S3×C6, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C6×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C6×C3⋊D4

Permutation representations of C6×C3⋊D4
►On 24 points - transitive group 24T248
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)

(1 20 15 11)(2 21 16 12)(3 22 17 7)(4 23 18 8)(5 24 13 9)(6 19 14 10)

(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)
 
G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,20,15,11)(2,21,16,12)(3,22,17,7)(4,23,18,8)(5,24,13,9)(6,19,14,10), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)>;
 
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), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,20,15,11)(2,21,16,12)(3,22,17,7)(4,23,18,8)(5,24,13,9)(6,19,14,10), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23) );
 
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)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,20,15,11),(2,21,16,12),(3,22,17,7),(4,23,18,8),(5,24,13,9),(6,19,14,10)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23)]])
 
G:=TransitiveGroup(24,248);
 

C6×C3⋊D4 is a maximal subgroup of
 C62.31D4  C62.100C23  C62.101C23  C62.56D4  C62.57D4  C62.111C23  C62.112C23  C62.113C23  C62.115C23  C62⋊4D4  C62⋊6D4  C62.121C23  C62⋊7D4  C62⋊8D4  C62.125C23  C32⋊2+ 1+4  S3×C6×D4

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A···6F6G···6AE6AF6AG6AH6AI12A12B12C12D
order1222222233333446···66···6666612121212
size1111226611222661···12···266666666

54 irreducible representations

dim111111111122222222
type++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4
kernelC6×C3⋊D4C6×Dic3C3×C3⋊D4S3×C2×C6C2×C62C2×C3⋊D4C2×Dic3C3⋊D4C22×S3C22×C6C22×C6C3×C6C2×C6C23C6C6C22C2
# reps114112282212324468

Matrix representation of C6×C3⋊D4 ►in GL3(𝔽13) generated by

400
0100
0010
,
100
030
009
,
100
001
0120
,
100
001
010
G:=sub<GL(3,GF(13))| [4,0,0,0,10,0,0,0,10],[1,0,0,0,3,0,0,0,9],[1,0,0,0,0,12,0,1,0],[1,0,0,0,0,1,0,1,0] >;
 

C6×C3⋊D4 in GAP, Magma, Sage, TeX 

C_6\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("C6xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(144,167);
 
// by ID
 
G=gap.SmallGroup(144,167);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,506,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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