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## G = C2×C62⋊S3order 432 = 24·33

### Direct product of C2 and C62⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C2×C62⋊S3
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32⋊A4 — C62⋊S3 — C2×C62⋊S3
 Lower central C3×A4 — C2×C62⋊S3
 Upper central C1 — C2

Generators and relations for C2×C62⋊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 [×4], C3, C3 [×3], C4 [×2], C22, C22 [×6], S3 [×4], C6, C6 [×11], C2×C4, D4 [×4], C23, C23, C32, C32 [×2], Dic3 [×2], C12 [×2], A4 [×2], D6 [×5], C2×C6, C2×C6 [×11], C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×4], S4 [×2], C2×A4 [×2], C22×S3, C22×C6, C22×C6 [×2], He3, C3×Dic3 [×2], C3×A4, C3×A4, S3×C6 [×4], C2×C3⋊S3, C62, C62 [×2], C2×C3⋊D4, C6×D4, C2×S4, C32⋊C6 [×2], C2×He3, C6×Dic3, C3×C3⋊D4 [×4], C3⋊S4 [×2], C6×A4, C6×A4, S3×C2×C6, C2×C62, C32⋊A4, C2×C32⋊C6, C6×C3⋊D4, C2×C3⋊S4, C62⋊S3 [×2], C2×C32⋊A4, C2×C62⋊S3
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C32⋊C6, C3×S4, C2×C32⋊C6, C6×S4, C62⋊S3, C2×C62⋊S3

Permutation representations of C2×C62⋊S3
On 18 points - transitive group 18T149
Generators in S18
(1 2)(3 4)(5 6)(7 10)(8 11)(9 12)(13 17)(14 18)(15 16)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 8 9)(10 11 12)(13 15 14)(16 18 17)
(1 2)(3 4)(5 6)(13 17)(14 18)(15 16)
(7 10)(8 11)(9 12)(13 17)(14 18)(15 16)
(1 17 9)(2 13 12)(3 18 8)(4 14 11)(5 16 7)(6 15 10)
(1 2)(3 6)(4 5)(7 14)(8 15)(9 13)(10 18)(11 16)(12 17)

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

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

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

G:=TransitiveGroup(18,149);

38 conjugacy classes

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

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 C32⋊C6 C2×C32⋊C6 C62⋊S3 C62⋊S3 C2×C62⋊S3 C2×C62⋊S3 kernel C2×C62⋊S3 C62⋊S3 C2×C32⋊A4 C2×C3⋊S4 C3⋊S4 C6×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×C62⋊S3 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 -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 -1 1 0 0 0 0 -1 0 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 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

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] >;

C2×C62⋊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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