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G = C2×C62⋊C6order 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), C625(C2×C6), (C2×C62)⋊3C6, C32⋊(C22×A4), C32⋊A44C22, C232(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, C222(C2×C32⋊C6), (C22×C6).16(C3×S3), SmallGroup(432,542)

Series: Derived Chief Lower central Upper central

C1C62 — C2×C62⋊C6
C1C3C32C62C32⋊A4C62⋊C6 — C2×C62⋊C6
C62 — C2×C62⋊C6
C1C2

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 [×6], C3, C3 [×3], C22, C22 [×12], S3 [×12], C6, C6 [×13], C23, C23 [×6], C32, C32 [×2], A4 [×2], D6 [×38], C2×C6, C2×C6 [×10], C24, C3×S3 [×2], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C2×A4 [×4], C22×S3 [×20], C22×C6, C22×C6, He3, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C2×C3⋊S3 [×9], C62, C62 [×2], C22×A4, S3×C23 [×2], C32⋊C6 [×2], C2×He3, S3×A4 [×2], C6×A4, C6×A4, C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C2×C62, C32⋊A4, C2×C32⋊C6, C2×S3×A4, C23×C3⋊S3, C62⋊C6 [×2], C2×C32⋊A4, C2×C62⋊C6
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], A4, D6, C2×C6, C3×S3, C2×A4 [×3], 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 3 5)(2 4 6)(7 12 8 10 9 11)(13 14 15 16 17 18)
(1 16 7 2 13 10)(3 18 9 6 17 11)(4 15 12 5 14 8)

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

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

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

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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