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G = C6xC3:D4  order 144 = 24·32

Direct product of C6 and C3:D4

direct product, metabelian, supersoluble, monomial

Aliases: C6xC3:D4, C62:7C22, (C2xC6):9D6, C3:3(C6xD4), (C3xC6):6D4, C6:2(C3xD4), D6:3(C2xC6), C22:4(S3xC6), (C2xC62):2C2, C23:3(C3xS3), (C22xC6):4C6, (C22xC6):3S3, C32:12(C2xD4), (C22xS3):4C6, (S3xC6):9C22, (C2xDic3):4C6, Dic3:2(C2xC6), (C6xDic3):10C2, C6.10(C22xC6), (C3xC6).28C23, C6.49(C22xS3), (C3xDic3):9C22, (S3xC2xC6):6C2, (C2xC6):5(C2xC6), C2.10(S3xC2xC6), SmallGroup(144,167)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C6xC3:D4
Chief series
C1 — C3 — C6 — C3xC6 — S3xC6 — S3xC2xC6 — C6xC3:D4
Lower central
C3 — C6 — C6xC3:D4
Upper central
C1 — C2xC6 — C22xC6

Generators and relations for C6xC3: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, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C2xD4, C3xS3, C3xC6, C3xC6, C3xC6, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C3xDic3, S3xC6, S3xC6, C62, C62, C62, C2xC3:D4, C6xD4, C6xDic3, C3xC3:D4, S3xC2xC6, C2xC62, C6xC3:D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3:D4, C3xD4, C22xS3, C22xC6, S3xC6, C2xC3:D4, C6xD4, C3xC3:D4, S3xC2xC6, C6xC3:D4

Permutation representations of C6xC3: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);
 

C6xC3: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  S3xC6xD4

54 conjugacy classes

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

54 irreducible representations

dim111111111122222222
type++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D6C3xS3C3:D4C3xD4S3xC6C3xC3:D4
kernelC6xC3:D4C6xDic3C3xC3:D4S3xC2xC6C2xC62C2xC3:D4C2xDic3C3:D4C22xS3C22xC6C22xC6C3xC6C2xC6C23C6C6C22C2
# reps114112282212324468

Matrix representation of C6xC3: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] >;
 

C6xC3: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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