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## G = C2×C32.S4order 432 = 24·33

### Direct product of C2 and C32.S4

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 [×4], C3, C3, C4 [×2], C22, C22 [×6], S3 [×2], C6, C6 [×9], C2×C4, D4 [×4], C23, C23, C9 [×2], C32, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×11], C2×D4, D9 [×2], C18 [×2], C3×S3 [×2], C3×C6, C3×C6 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×4], C22×S3, C22×C6, C22×C6 [×2], 3- 1+2, C3.A4, C3.A4, D18, C3×Dic3 [×2], S3×C6 [×4], C62, C62 [×2], C2×C3⋊D4, C6×D4, C9⋊C6 [×2], C2×3- 1+2, C3.S4 [×2], C2×C3.A4, C2×C3.A4, C6×Dic3, C3×C3⋊D4 [×4], S3×C2×C6, C2×C62, C32.A4, C2×C9⋊C6, C2×C3.S4, C6×C3⋊D4, C32.S4 [×2], C2×C32.A4, C2×C32.S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], 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 18)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)
(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 18)(2 10)(4 12)(5 13)(7 15)(8 16)
(2 10)(3 11)(5 13)(6 14)(8 16)(9 17)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)

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

G:=Group( (1,18)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17), (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), (1,18)(2,10)(4,12)(5,13)(7,15)(8,16), (2,10)(3,11)(5,13)(6,14)(8,16)(9,17), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10) );

G=PermutationGroup([(1,18),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17)], [(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)], [(1,18),(2,10),(4,12),(5,13),(7,15),(8,16)], [(2,10),(3,11),(5,13),(6,14),(8,16),(9,17)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)])

G:=TransitiveGroup(18,147);

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 6A 6B ··· 6G 6H ··· 6M 6N 6O 6P 6Q 9A 9B 9C 12A 12B 12C 12D 18A 18B 18C order 1 2 2 2 2 2 3 3 3 4 4 6 6 ··· 6 6 ··· 6 6 6 6 6 9 9 9 12 12 12 12 18 18 18 size 1 1 3 3 18 18 2 3 3 18 18 2 3 ··· 3 6 ··· 6 18 18 18 18 24 24 24 18 18 18 18 24 24 24

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 S4 C2×S4 C3×S4 C6×S4 C9⋊C6 C2×C9⋊C6 C32.S4 C32.S4 C2×C32.S4 C2×C32.S4 kernel C2×C32.S4 C32.S4 C2×C32.A4 C2×C3.S4 C3.S4 C2×C3.A4 C2×C62 C62 C22×C6 C2×C6 C3×C6 C32 C6 C3 C23 C22 C2 C2 C1 C1 # reps 1 2 1 2 4 2 1 1 2 2 2 2 4 4 1 1 1 2 1 2

Matrix representation of C2×C32.S4 in GL6(ℤ)

 -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 0 1 0 0 0 0 0 0 1 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1 0 0 -1 0 0 0 0 0 -1 1 0 0

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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