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G = C2xC62:S3  order 432 = 24·33

Direct product of C2 and C62:S3

direct product, non-abelian, soluble, monomial

Aliases: C2xC62:S3, C62:14D6, C3:S4:C6, (C6xA4):C6, (C3xC6):1S4, C3.3(C6xS4), C6.12(C3xS4), C32:2(C2xS4), (C2xC62):4S3, C23:(C32:C6), C32:A4:2C22, (C2xC3:S4):C3, (C3xA4):(C2xC6), (C2xC6).4(S3xC6), C22:(C2xC32:C6), (C2xC32:A4):1C2, (C22xC6).9(C3xS3), SmallGroup(432,535)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3xA4 — C2xC62:S3
Chief series
C1 — C22 — C2xC6 — C3xA4 — C32:A4 — C62:S3 — C2xC62:S3
Lower central
C3xA4 — C2xC62:S3
Upper central
C1 — C2

Generators and relations for C2xC62:S3
 G = < a,b,c,d,e,f,g | a2=b3=c3=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, fbf-1=bc-1, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 847 in 134 conjugacy classes, 22 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, C32, Dic3, C12, A4, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, C2xDic3, C3:D4, C2xC12, C3xD4, S4, C2xA4, C22xS3, C22xC6, C22xC6, He3, C3xDic3, C3xA4, C3xA4, S3xC6, C2xC3:S3, C62, C62, C2xC3:D4, C6xD4, C2xS4, C32:C6, C2xHe3, C6xDic3, C3xC3:D4, C3:S4, C6xA4, C6xA4, S3xC2xC6, C2xC62, C32:A4, C2xC32:C6, C6xC3:D4, C2xC3:S4, C62:S3, C2xC32:A4, C2xC62:S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, S4, S3xC6, C2xS4, C32:C6, C3xS4, C2xC32:C6, C6xS4, C62:S3, C2xC62:S3

Permutation representations of C2xC62:S3
►On 18 points - transitive group 18T149
Generators in S18
(1 2)(3 4)(5 6)(7 13)(8 14)(9 15)(10 18)(11 16)(12 17)

(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 5 3)(2 6 4)(7 8 9)(10 12 11)(13 14 15)(16 18 17)

(1 2)(3 4)(5 6)(10 18)(11 16)(12 17)

(7 13)(8 14)(9 15)(10 18)(11 16)(12 17)

(1 16 9)(2 11 15)(3 17 8)(4 12 14)(5 18 7)(6 10 13)

(1 2)(3 6)(4 5)(7 12)(8 10)(9 11)(13 17)(14 18)(15 16)
 
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,8,9)(10,12,11)(13,14,15)(16,18,17), (1,2)(3,4)(5,6)(10,18)(11,16)(12,17), (7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (1,16,9)(2,11,15)(3,17,8)(4,12,14)(5,18,7)(6,10,13), (1,2)(3,6)(4,5)(7,12)(8,10)(9,11)(13,17)(14,18)(15,16)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,3)(2,6,4)(7,8,9)(10,12,11)(13,14,15)(16,18,17), (1,2)(3,4)(5,6)(10,18)(11,16)(12,17), (7,13)(8,14)(9,15)(10,18)(11,16)(12,17), (1,16,9)(2,11,15)(3,17,8)(4,12,14)(5,18,7)(6,10,13), (1,2)(3,6)(4,5)(7,12)(8,10)(9,11)(13,17)(14,18)(15,16) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,13),(8,14),(9,15),(10,18),(11,16),(12,17)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,3),(2,6,4),(7,8,9),(10,12,11),(13,14,15),(16,18,17)], [(1,2),(3,4),(5,6),(10,18),(11,16),(12,17)], [(7,13),(8,14),(9,15),(10,18),(11,16),(12,17)], [(1,16,9),(2,11,15),(3,17,8),(4,12,14),(5,18,7),(6,10,13)], [(1,2),(3,6),(4,5),(7,12),(8,10),(9,11),(13,17),(14,18),(15,16)]])
 
G:=TransitiveGroup(18,149);
 

38 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B6A6B···6G6H···6M6N6O6P6Q6R6S6T12A12B12C12D
order1222223333334466···66···6666666612121212
size11331818233242424181823···36···61818181824242418181818

38 irreducible representations

dim11111122223333666666
type+++++++++++
imageC1C2C2C3C6C6S3D6C3xS3S3xC6S4C2xS4C3xS4C6xS4C32:C6C2xC32:C6C62:S3C62:S3C2xC62:S3C2xC62:S3
kernelC2xC62:S3C62:S3C2xC32:A4C2xC3:S4C3:S4C6xA4C2xC62C62C22xC6C2xC6C3xC6C32C6C3C23C22C2C2C1C1
# reps12124211222244111212

Matrix representation of C2xC62:S3 ►in GL6(Z)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
100000
010000
00-1100
00-1000
00000-1
00001-1
,
-110000
-100000
00-1100
00-1000
0000-11
0000-10
,
-100000
0-10000
001000
000100
0000-10
00000-1
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
00-1100
00-1000
0000-11
0000-10
-110000
-100000
,
010000
100000
000001
000010
000100
001000

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,0,0,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
 

C2xC62:S3 in GAP, Magma, Sage, TeX 

C_2\times C_6^2\rtimes S_3
 
% in TeX
 
G:=Group("C2xC6^2:S3");
 
// GroupNames label
 
G:=SmallGroup(432,535);
 
// by ID
 
G=gap.SmallGroup(432,535);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,675,353,2524,9077,782,5298,1350]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^3=c^3=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c^-1,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g=c^-1,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
 
// generators/relations
 

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