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G = C2xC32.S4  order 432 = 24·33

Direct product of C2 and C32.S4

direct product, non-abelian, soluble, monomial

Aliases: C2xC32.S4, C62.49D6, C3.S4:C6, C23:(C9:C6), C3.1(C6xS4), C32.(C2xS4), C6.10(C3xS4), (C3xC6).10S4, C32.A4:C22, (C2xC62).14S3, (C2xC3.S4):C3, (C2xC3.A4):C6, C3.A4:(C2xC6), C22:(C2xC9:C6), (C2xC6).2(S3xC6), (C2xC32.A4):C2, (C22xC6).7(C3xS3), SmallGroup(432,533)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3.A4 — C2xC32.S4
Chief series
C1 — C22 — C2xC6 — C3.A4 — C32.A4 — C32.S4 — C2xC32.S4
Lower central
C3.A4 — C2xC32.S4
Upper central
C1 — C2

Generators and relations for C2xC32.S4
 G = < a,b,c,d,e,f,g | a2=b3=c3=d2=e2=g2=1, f3=c, 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=c-1f2 >

Subgroups: 622 in 120 conjugacy classes, 22 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C9, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xD4, D9, C18, C3xS3, C3xC6, C3xC6, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, 3- 1+2, C3.A4, C3.A4, D18, C3xDic3, S3xC6, C62, C62, C2xC3:D4, C6xD4, C9:C6, C2x3- 1+2, C3.S4, C2xC3.A4, C2xC3.A4, C6xDic3, C3xC3:D4, S3xC2xC6, C2xC62, C32.A4, C2xC9:C6, C2xC3.S4, C6xC3:D4, C32.S4, C2xC32.A4, C2xC32.S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, S4, S3xC6, C2xS4, C9:C6, C3xS4, C2xC9:C6, C6xS4, C32.S4, C2xC32.S4

Permutation representations of C2xC32.S4
►On 18 points - transitive group 18T147
Generators in S18
(1 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)

(2 8 5)(3 6 9)(10 16 13)(11 14 17)

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

(1 15)(2 16)(4 18)(5 10)(7 12)(8 13)

(2 16)(3 17)(5 10)(6 11)(8 13)(9 14)

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

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 18)(8 17)(9 16)
 
G:=sub<Sym(18)| (1,15)(2,16)(3,17)(4,18)(5,10)(6,11)(7,12)(8,13)(9,14), (2,8,5)(3,6,9)(10,16,13)(11,14,17), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,15)(2,16)(4,18)(5,10)(7,12)(8,13), (2,16)(3,17)(5,10)(6,11)(8,13)(9,14), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,18)(8,17)(9,16)>;
 
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,10)(6,11)(7,12)(8,13)(9,14), (2,8,5)(3,6,9)(10,16,13)(11,14,17), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,15)(2,16)(4,18)(5,10)(7,12)(8,13), (2,16)(3,17)(5,10)(6,11)(8,13)(9,14), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,18)(8,17)(9,16) );
 
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,10),(6,11),(7,12),(8,13),(9,14)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,15),(2,16),(4,18),(5,10),(7,12),(8,13)], [(2,16),(3,17),(5,10),(6,11),(8,13),(9,14)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,18),(8,17),(9,16)]])
 
G:=TransitiveGroup(18,147);
 

38 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B6A6B···6G6H···6M6N6O6P6Q9A9B9C12A12B12C12D18A18B18C
order1222223334466···66···6666699912121212181818
size11331818233181823···36···61818181824242418181818242424

38 irreducible representations

dim11111122223333666666
type+++++++++++
imageC1C2C2C3C6C6S3D6C3xS3S3xC6S4C2xS4C3xS4C6xS4C9:C6C2xC9:C6C32.S4C32.S4C2xC32.S4C2xC32.S4
kernelC2xC32.S4C32.S4C2xC32.A4C2xC3.S4C3.S4C2xC3.A4C2xC62C62C22xC6C2xC6C3xC6C32C6C3C23C22C2C2C1C1
# reps12124211222244111212

Matrix representation of C2xC32.S4 ►in GL6(Z)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
100000
010000
000-100
001-100
0000-11
0000-10
,
-110000
-100000
00-1100
00-1000
0000-11
0000-10
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
000010
000001
0-10000
1-10000
000-100
001-100
,
010000
100000
0000-10
0000-11
00-1000
00-1100

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-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],[0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,1,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,-1,-1,0,0,0,0,0,1,0,0] >;
 

C2xC32.S4 in GAP, Magma, Sage, TeX 

C_2\times C_3^2.S_4
 
% in TeX
 
G:=Group("C2xC3^2.S4");
 
// GroupNames label
 
G:=SmallGroup(432,533);
 
// by ID
 
G=gap.SmallGroup(432,533);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,1683,353,192,2524,9077,782,5298,1350]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^3=c^3=d^2=e^2=g^2=1,f^3=c,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=c^-1*f^2>;
 
// generators/relations
 

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