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G = C2×C32.S4  order 432 = 24·33

Direct product of C2 and C32.S4

direct product, non-abelian, soluble, monomial

Aliases: C2×C32.S4, C62.49D6, C3.S4⋊C6, C23⋊(C9⋊C6), C3.1(C6×S4), C32.(C2×S4), C6.10(C3×S4), (C3×C6).10S4, C32.A4⋊C22, (C2×C62).14S3, (C2×C3.S4)⋊C3, (C2×C3.A4)⋊C6, C3.A4⋊(C2×C6), C22⋊(C2×C9⋊C6), (C2×C6).2(S3×C6), (C2×C32.A4)⋊C2, (C22×C6).7(C3×S3), SmallGroup(432,533)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3.A4 — C2×C32.S4
Chief series
C1 — C22 — C2×C6 — C3.A4 — C32.A4 — C32.S4 — C2×C32.S4
Lower central
C3.A4 — C2×C32.S4
Upper central
C1 — C2

Generators and relations for C2×C32.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, C2×C4, D4, C23, C23, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C3×S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, 3- 1+2, C3.A4, C3.A4, D18, C3×Dic3, S3×C6, C62, C62, C2×C3⋊D4, C6×D4, C9⋊C6, C2×3- 1+2, C3.S4, C2×C3.A4, C2×C3.A4, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C32.A4, C2×C9⋊C6, C2×C3.S4, C6×C3⋊D4, C32.S4, C2×C32.A4, C2×C32.S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C9⋊C6, C3×S4, C2×C9⋊C6, C6×S4, C32.S4, C2×C32.S4

Permutation representations of C2×C32.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+++++++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6S4C2×S4C3×S4C6×S4C9⋊C6C2×C9⋊C6C32.S4C32.S4C2×C32.S4C2×C32.S4
kernelC2×C32.S4C32.S4C2×C32.A4C2×C3.S4C3.S4C2×C3.A4C2×C62C62C22×C6C2×C6C3×C6C32C6C3C23C22C2C2C1C1
# reps12124211222244111212

Matrix representation of C2×C32.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] >;
 

C2×C32.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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