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## G = C2×C33⋊C6order 324 = 22·34

### Direct product of C2 and C33⋊C6

Aliases: C2×C33⋊C6, He32D6, C3≀C34C22, (C2×He3)⋊1S3, C332(C2×C6), (C32×C6)⋊1C6, C33⋊C22C6, C32.5(S3×C6), C6.6(C32⋊C6), (C2×C3≀C3)⋊3C2, (C3×C6).16(C3×S3), C3.2(C2×C32⋊C6), (C2×C33⋊C2)⋊1C3, SmallGroup(324,69)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C33⋊C6
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C33⋊C6 — C2×C33⋊C6
 Lower central C33 — C2×C33⋊C6
 Upper central C1 — C2

Generators and relations for C2×C33⋊C6
G = < a,b,c,d,e | a2=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1c-1, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 772 in 84 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×5], C22, S3 [×10], C6, C6 [×7], C9, C32, C32 [×5], D6 [×5], C2×C6, C18, C3×S3 [×2], C3⋊S3 [×10], C3×C6, C3×C6 [×5], He3, 3- 1+2, C33, S3×C6, C2×C3⋊S3 [×5], C32⋊C6 [×2], C2×He3, C2×3- 1+2, C33⋊C2 [×2], C32×C6, C3≀C3, C2×C32⋊C6, C2×C33⋊C2, C33⋊C6 [×2], C2×C3≀C3, C2×C33⋊C6
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, C33⋊C6, C2×C33⋊C6

Character table of C2×C33⋊C6

 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 6 6 6 6 9 9 2 6 6 6 6 9 9 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 1 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 -1 -1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ7 1 -1 1 -1 1 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 -1 ζ65 ζ6 ζ32 ζ6 ζ3 ζ65 ζ32 ζ3 ζ6 ζ65 linear of order 6 ρ8 1 -1 -1 1 1 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 -1 ζ65 ζ6 ζ6 ζ32 ζ65 ζ3 ζ32 ζ3 ζ6 ζ65 linear of order 6 ρ9 1 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ10 1 -1 1 -1 1 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 -1 ζ6 ζ65 ζ3 ζ65 ζ32 ζ6 ζ3 ζ32 ζ65 ζ6 linear of order 6 ρ11 1 1 -1 -1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ12 1 -1 -1 1 1 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 -1 ζ6 ζ65 ζ65 ζ3 ζ6 ζ32 ζ3 ζ32 ζ65 ζ6 linear of order 6 ρ13 2 -2 0 0 2 -1 -1 2 -1 2 2 -2 1 1 -2 1 -2 -2 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ14 2 2 0 0 2 -1 -1 2 -1 2 2 2 -1 -1 2 -1 2 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 -2 0 0 2 -1 -1 2 -1 -1+√-3 -1-√-3 -2 1 1 -2 1 1+√-3 1-√-3 0 0 0 0 ζ65 ζ6 ζ3 ζ32 complex lifted from S3×C6 ρ16 2 2 0 0 2 -1 -1 2 -1 -1+√-3 -1-√-3 2 -1 -1 2 -1 -1-√-3 -1+√-3 0 0 0 0 ζ65 ζ6 ζ65 ζ6 complex lifted from C3×S3 ρ17 2 -2 0 0 2 -1 -1 2 -1 -1-√-3 -1+√-3 -2 1 1 -2 1 1-√-3 1+√-3 0 0 0 0 ζ6 ζ65 ζ32 ζ3 complex lifted from S3×C6 ρ18 2 2 0 0 2 -1 -1 2 -1 -1-√-3 -1+√-3 2 -1 -1 2 -1 -1+√-3 -1-√-3 0 0 0 0 ζ6 ζ65 ζ6 ζ65 complex lifted from C3×S3 ρ19 6 -6 0 0 -3 3 -3 0 0 0 0 3 0 3 0 -3 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ20 6 6 0 0 6 0 0 -3 0 0 0 6 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ21 6 6 0 0 -3 3 -3 0 0 0 0 -3 0 -3 0 3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊C6 ρ22 6 6 0 0 -3 -3 0 0 3 0 0 -3 3 0 0 -3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊C6 ρ23 6 6 0 0 -3 0 3 0 -3 0 0 -3 -3 3 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊C6 ρ24 6 -6 0 0 -3 -3 0 0 3 0 0 3 -3 0 0 3 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 6 -6 0 0 6 0 0 -3 0 0 0 -6 0 0 3 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C6 ρ26 6 -6 0 0 -3 0 3 0 -3 0 0 3 3 -3 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C2×C33⋊C6
On 18 points - transitive group 18T125
Generators in S18
(1 4)(2 5)(3 6)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(2 16 7)(3 8 17)(5 10 13)(6 14 11)
(1 12 15)(3 17 8)(4 18 9)(6 11 14)
(1 15 12)(2 7 16)(3 17 8)(4 9 18)(5 13 10)(6 11 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,125);

Matrix representation of C2×C33⋊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
,
 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0
,
 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 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 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0
,
 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],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[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×C33⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes C_6
% in TeX

G:=Group("C2xC3^3:C6");
// GroupNames label

G:=SmallGroup(324,69);
// by ID

G=gap.SmallGroup(324,69);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,303,2164,1096,7781]);
// Polycyclic

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

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