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## G = C2×C62⋊C6order 432 = 24·33

### Direct product of C2 and C62⋊C6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C62⋊C6
G = < a,b,c,d | a2=b6=c6=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b3c2 >

Subgroups: 1543 in 192 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, C32, A4, D6, C2×C6, C2×C6, C24, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C2×A4, C22×S3, C22×C6, C22×C6, He3, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C22×A4, S3×C23, C32⋊C6, C2×He3, S3×A4, C6×A4, C6×A4, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C32⋊A4, C2×C32⋊C6, C2×S3×A4, C23×C3⋊S3, C62⋊C6, C2×C32⋊A4, C2×C62⋊C6
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, S3×C6, C22×A4, C32⋊C6, S3×A4, C2×C32⋊C6, C2×S3×A4, C62⋊C6, C2×C62⋊C6

Permutation representations of C2×C62⋊C6
On 18 points - transitive group 18T148
Generators in S18
(1 2)(3 4)(5 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5)(2 3 6)(7 12 8 10 9 11)(13 14 15 16 17 18)
(1 15 7 2 18 10)(3 14 12 5 13 8)(4 17 9 6 16 11)

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

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

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

G:=TransitiveGroup(18,148);

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 6A 6B ··· 6J 6K 6L 6M 6N 6O 6P 6Q 6R order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 6 6 ··· 6 6 6 6 6 6 6 6 6 size 1 1 3 3 9 9 27 27 2 6 12 12 24 24 2 6 ··· 6 12 12 24 24 36 36 36 36

32 irreducible representations

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

Matrix representation of C2×C62⋊C6 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
,
 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 0 0 0 0 -1 0 0 0 0 1 1
,
 0 0 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 1 0 0 0 0 0 -1 -1 0 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],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,1],[0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0] >;

C2×C62⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_6^2\rtimes C_6
% in TeX

G:=Group("C2xC6^2:C6");
// GroupNames label

G:=SmallGroup(432,542);
// by ID

G=gap.SmallGroup(432,542);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-3,269,123,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=b^3*c^2>;
// generators/relations

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