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## G = C2×C32⋊A4order 216 = 23·33

### Direct product of C2 and C32⋊A4

Aliases: C2×C32⋊A4, C23⋊He3, C627C6, (C3×C6)⋊A4, (C6×A4)⋊C3, (C3×A4)⋊2C6, C6.7(C3×A4), C3.5(C6×A4), C22⋊(C2×He3), (C2×C62)⋊1C3, C322(C2×A4), (C22×C6).5C32, (C2×C6).5(C3×C6), SmallGroup(216,107)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C32⋊A4
G = < a,b,c,d,e,f | a2=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 226 in 68 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C22, C22 [×2], C6, C6 [×12], C23, C32, C32 [×3], A4 [×3], C2×C6, C2×C6 [×9], C3×C6, C3×C6 [×5], C2×A4 [×3], C22×C6, C22×C6, He3, C3×A4 [×3], C62, C62 [×2], C2×He3, C6×A4 [×3], C2×C62, C32⋊A4, C2×C32⋊A4
Quotients: C1, C2, C3 [×4], C6 [×4], C32, A4, C3×C6, C2×A4, He3, C3×A4, C2×He3, C6×A4, C32⋊A4, C2×C32⋊A4

Permutation representations of C2×C32⋊A4
On 18 points - transitive group 18T91
Generators in S18
(1 2)(3 4)(5 6)(7 10)(8 11)(9 12)(13 18)(14 16)(15 17)
(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)
(7 10)(8 11)(9 12)(13 18)(14 16)(15 17)
(1 2)(3 4)(5 6)(7 10)(8 11)(9 12)
(1 13 7)(2 18 10)(3 14 9)(4 16 12)(5 15 8)(6 17 11)

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

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

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

G:=TransitiveGroup(18,91);

C2×C32⋊A4 is a maximal subgroup of   C625Dic3  C626Dic3  C624C12
C2×C32⋊A4 is a maximal quotient of   C4○D4⋊He3

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E ··· 3J 6A 6B 6C ··· 6T 6U ··· 6Z order 1 2 2 2 3 3 3 3 3 ··· 3 6 6 6 ··· 6 6 ··· 6 size 1 1 3 3 1 1 3 3 12 ··· 12 1 1 3 ··· 3 12 ··· 12

40 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 He3 C3×A4 C2×He3 C6×A4 C32⋊A4 C2×C32⋊A4 kernel C2×C32⋊A4 C32⋊A4 C6×A4 C2×C62 C3×A4 C62 C3×C6 C32 C23 C6 C22 C3 C2 C1 # reps 1 1 6 2 6 2 1 1 2 2 2 2 6 6

Matrix representation of C2×C32⋊A4 in GL3(𝔽7) generated by

 6 0 0 0 6 0 0 0 6
,
 1 0 0 0 4 0 0 0 2
,
 4 0 0 0 4 0 0 0 4
,
 6 0 0 0 6 0 0 0 1
,
 6 0 0 0 1 0 0 0 6
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[1,0,0,0,4,0,0,0,2],[4,0,0,0,4,0,0,0,4],[6,0,0,0,6,0,0,0,1],[6,0,0,0,1,0,0,0,6],[0,0,1,1,0,0,0,1,0] >;

C2×C32⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes A_4
% in TeX

G:=Group("C2xC3^2:A4");
// GroupNames label

G:=SmallGroup(216,107);
// by ID

G=gap.SmallGroup(216,107);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,224,1630,2927]);
// Polycyclic

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

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