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G = C2×C62⋊C6  order 432 = 24·33

Direct product of C2 and C62⋊C6

direct product, metabelian, soluble, monomial

Aliases: C2×C62⋊C6, (C6×A4)⋊2S3, (C3×A4)⋊3D6, C6.12(S3×A4), C62⋊5(C2×C6), (C2×C62)⋊3C6, C32⋊(C22×A4), C32⋊A4⋊4C22, C23⋊2(C32⋊C6), C3⋊S3⋊(C2×A4), (C2×C3⋊S3)⋊A4, (C3×C6)⋊(C2×A4), C3.3(C2×S3×A4), (C2×C6).7(S3×C6), (C2×C32⋊A4)⋊3C2, (C23×C3⋊S3)⋊1C3, (C22×C3⋊S3)⋊5C6, C22⋊2(C2×C32⋊C6), (C22×C6).16(C3×S3), SmallGroup(432,542)

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 2A2B2C2D2E2F2G3A3B3C3D3E3F6A6B···6J6K6L6M6N6O6P6Q6R
order1222222233333366···666666666
size1133992727261212242426···61212242436363636

32 irreducible representations

dim1111112222333666666
type++++++++++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6A4C2×A4C2×A4C32⋊C6S3×A4C2×C32⋊C6C2×S3×A4C62⋊C6C2×C62⋊C6
kernelC2×C62⋊C6C62⋊C6C2×C32⋊A4C23×C3⋊S3C22×C3⋊S3C2×C62C6×A4C3×A4C22×C6C2×C6C2×C3⋊S3C3⋊S3C3×C6C23C6C22C3C2C1
# reps1212421122121111133

Matrix representation of C2×C62⋊C6 ►in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
00-1-100
001000
00000-1
000011
,
010000
-1-10000
000-100
001100
00000-1
000011
,
001000
00-1-100
000010
0000-1-1
100000
-1-10000

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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