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## G = C2×C33⋊S3order 324 = 22·34

### Direct product of C2 and C33⋊S3

Aliases: C2×C33⋊S3, He33D6, C338D6, 3- 1+21D6, C3≀C32C22, (C2×He3)⋊2S3, (C32×C6)⋊2S3, C6.7(He3⋊C2), (C2×3- 1+2)⋊1S3, (C2×C3≀C3)⋊1C2, (C3×C6).5(C3⋊S3), C32.1(C2×C3⋊S3), C3.2(C2×He3⋊C2), SmallGroup(324,77)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3≀C3 — C2×C33⋊S3
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C33⋊S3 — C2×C33⋊S3
 Lower central C3≀C3 — C2×C33⋊S3
 Upper central C1 — C2

Generators and relations for C2×C33⋊S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=b-1, ece-1=cd=dc, fcf=cd-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 634 in 100 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×5], C22, S3 [×10], C6, C6 [×7], C9 [×2], C32, C32 [×5], D6 [×5], C2×C6, D9 [×4], C18 [×2], C3×S3 [×8], C3⋊S3 [×4], C3×C6, C3×C6 [×5], He3, 3- 1+2 [×2], C33, D18 [×2], S3×C6 [×4], C2×C3⋊S3 [×2], C32⋊C6 [×2], C9⋊C6 [×4], C2×He3, C2×3- 1+2 [×2], C3×C3⋊S3 [×2], C32×C6, C3≀C3, C2×C32⋊C6, C2×C9⋊C6 [×2], C6×C3⋊S3, C33⋊S3 [×2], C2×C3≀C3, C2×C33⋊S3
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, C33⋊S3, C2×C33⋊S3

Character table of C2×C33⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 9A 9B 18A 18B size 1 1 27 27 2 3 3 6 6 6 18 2 3 3 6 6 6 18 27 27 27 27 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 2 2 2 -1 -1 -1 2 -2 -2 -2 1 1 1 -2 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ7 2 -2 0 0 2 2 2 -1 -1 -1 -1 -2 -2 -2 1 1 1 1 0 0 0 0 2 -1 -2 1 orthogonal lifted from D6 ρ8 2 2 0 0 2 2 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 0 0 0 0 2 -1 2 -1 orthogonal lifted from S3 ρ9 2 -2 0 0 2 2 2 -1 -1 -1 -1 -2 -2 -2 1 1 1 1 0 0 0 0 -1 2 1 -2 orthogonal lifted from D6 ρ10 2 2 0 0 2 2 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 0 0 0 0 -1 2 -1 2 orthogonal lifted from S3 ρ11 2 2 0 0 2 2 2 2 2 2 -1 2 2 2 2 2 2 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 0 0 2 2 2 2 2 2 -1 -2 -2 -2 -2 -2 -2 1 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ13 3 3 1 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ32 ζ3 ζ3 0 0 0 0 complex lifted from He3⋊C2 ρ14 3 3 -1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ6 ζ65 ζ65 0 0 0 0 complex lifted from He3⋊C2 ρ15 3 -3 1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -3 3+3√-3/2 3-3√-3/2 0 0 0 0 ζ32 ζ6 ζ3 ζ65 0 0 0 0 complex lifted from C2×He3⋊C2 ρ16 3 -3 -1 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -3 3+3√-3/2 3-3√-3/2 0 0 0 0 ζ6 ζ32 ζ65 ζ3 0 0 0 0 complex lifted from C2×He3⋊C2 ρ17 3 3 -1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ65 ζ6 ζ6 0 0 0 0 complex lifted from He3⋊C2 ρ18 3 -3 1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -3 3-3√-3/2 3+3√-3/2 0 0 0 0 ζ3 ζ65 ζ32 ζ6 0 0 0 0 complex lifted from C2×He3⋊C2 ρ19 3 3 1 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ3 ζ32 ζ32 0 0 0 0 complex lifted from He3⋊C2 ρ20 3 -3 -1 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -3 3-3√-3/2 3+3√-3/2 0 0 0 0 ζ65 ζ3 ζ6 ζ32 0 0 0 0 complex lifted from C2×He3⋊C2 ρ21 6 -6 0 0 -3 0 0 -3 0 3 0 3 0 0 -3 3 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ22 6 6 0 0 -3 0 0 -3 0 3 0 -3 0 0 3 -3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ23 6 -6 0 0 -3 0 0 0 3 -3 0 3 0 0 3 0 -3 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 6 -6 0 0 -3 0 0 3 -3 0 0 3 0 0 0 -3 3 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 6 6 0 0 -3 0 0 0 3 -3 0 -3 0 0 -3 0 3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ26 6 6 0 0 -3 0 0 3 -3 0 0 -3 0 0 0 3 -3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3

Permutation representations of C2×C33⋊S3
On 18 points - transitive group 18T134
Generators in S18
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 2 3)(7 8 9)(10 11 12)(16 17 18)
(1 3 2)(4 5 6)(7 8 9)(10 12 11)(13 14 15)(16 17 18)
(1 5 8)(2 4 7)(3 6 9)(10 14 17)(11 13 16)(12 15 18)
(1 10)(2 12)(3 11)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)

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

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

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

G:=TransitiveGroup(18,134);

Matrix representation of C2×C33⋊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
,
 0 1 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
,
 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 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
,
 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 0 0 1 0 0 0 0 0 -1 -1 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],[0,-1,0,0,0,0,1,-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],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-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],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0] >;

C2×C33⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes S_3
% in TeX

G:=Group("C2xC3^3:S3");
// GroupNames label

G:=SmallGroup(324,77);
// by ID

G=gap.SmallGroup(324,77);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,579,303,7564,1096,7781]);
// Polycyclic

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

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