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## G = C2×C32⋊D6order 216 = 23·33

### Direct product of C2 and C32⋊D6

Aliases: C2×C32⋊D6, He3⋊C23, C3⋊S3⋊D6, (C3×C6)⋊D6, C6.21S32, C32⋊C6⋊C22, (C2×He3)⋊C22, C32⋊(C22×S3), He3⋊C2⋊C22, C3.2(C2×S32), (C2×C3⋊S3)⋊3S3, (C2×C32⋊C6)⋊5C2, (C2×He3⋊C2)⋊4C2, SmallGroup(216,102)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×C32⋊D6
 Chief series C1 — C3 — C32 — He3 — C32⋊C6 — C32⋊D6 — C2×C32⋊D6
 Lower central He3 — C2×C32⋊D6
 Upper central C1 — C2

Generators and relations for C2×C32⋊D6
G = < a,b,c,d,e | a2=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1c-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 682 in 122 conjugacy classes, 30 normal (8 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C22 [×7], S3 [×14], C6, C6 [×9], C23, C32 [×2], C32, D6 [×19], C2×C6 [×3], C3×S3 [×10], C3⋊S3 [×4], C3×C6 [×2], C3×C6, C22×S3 [×3], He3, S32 [×8], S3×C6 [×5], C2×C3⋊S3 [×2], C32⋊C6 [×4], He3⋊C2 [×2], C2×He3, C2×S32 [×2], C32⋊D6 [×4], C2×C32⋊C6 [×2], C2×He3⋊C2, C2×C32⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C22×S3 [×2], S32, C2×S32, C32⋊D6, C2×C32⋊D6

Character table of C2×C32⋊D6

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J size 1 1 9 9 9 9 9 9 2 6 6 12 2 6 6 12 18 18 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 trivial ρ2 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 0 -2 0 0 0 2 2 2 -1 -1 -2 -2 1 1 0 0 -1 0 0 1 orthogonal lifted from D6 ρ10 2 2 0 -2 0 0 0 -2 2 2 -1 -1 2 2 -1 -1 0 0 1 0 0 1 orthogonal lifted from D6 ρ11 2 -2 0 0 -2 2 0 0 2 -1 2 -1 -2 1 -2 1 0 -1 0 0 1 0 orthogonal lifted from D6 ρ12 2 -2 0 0 2 -2 0 0 2 -1 2 -1 -2 1 -2 1 0 1 0 0 -1 0 orthogonal lifted from D6 ρ13 2 2 0 2 0 0 0 2 2 2 -1 -1 2 2 -1 -1 0 0 -1 0 0 -1 orthogonal lifted from S3 ρ14 2 -2 0 2 0 0 0 -2 2 2 -1 -1 -2 -2 1 1 0 0 1 0 0 -1 orthogonal lifted from D6 ρ15 2 2 0 0 2 2 0 0 2 -1 2 -1 2 -1 2 -1 0 -1 0 0 -1 0 orthogonal lifted from S3 ρ16 2 2 0 0 -2 -2 0 0 2 -1 2 -1 2 -1 2 -1 0 1 0 0 1 0 orthogonal lifted from D6 ρ17 4 4 0 0 0 0 0 0 4 -2 -2 1 4 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ18 4 -4 0 0 0 0 0 0 4 -2 -2 1 -4 2 2 -1 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ19 6 -6 -2 0 0 0 2 0 -3 0 0 0 3 0 0 0 1 0 0 -1 0 0 orthogonal faithful ρ20 6 -6 2 0 0 0 -2 0 -3 0 0 0 3 0 0 0 -1 0 0 1 0 0 orthogonal faithful ρ21 6 6 2 0 0 0 2 0 -3 0 0 0 -3 0 0 0 -1 0 0 -1 0 0 orthogonal lifted from C32⋊D6 ρ22 6 6 -2 0 0 0 -2 0 -3 0 0 0 -3 0 0 0 1 0 0 1 0 0 orthogonal lifted from C32⋊D6

Permutation representations of C2×C32⋊D6
On 18 points - transitive group 18T94
Generators in S18
(1 2)(3 6)(4 5)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(1 11 8)(2 17 14)(3 13 18)(4 15 16)(5 9 10)(6 7 12)
(1 6 5)(2 3 4)(7 9 11)(8 12 10)(13 15 17)(14 18 16)
(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(7 8)(9 12)(10 11)(13 14)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,94);

C2×C32⋊D6 is a maximal subgroup of
C32⋊D6⋊C4  C3⋊S3⋊D12  C12.86S32  C62⋊D6  C622D6
C2×C32⋊D6 is a maximal quotient of
C3⋊S3⋊Dic6  C12⋊S3⋊S3  C12.84S32  C12.91S32  C12.85S32  C12.S32  C3⋊S3⋊D12  C12.86S32  C62.8D6  C62.9D6  C62⋊D6  C622D6

Matrix representation of C2×C32⋊D6 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 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
,
 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
,
 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 1 1 0 0 0 0 0 -1 0 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 -1 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],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[0,-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],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,-1,0,0,0,-1,0,0,0,0,-1,0,0,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,-1,1,0,0] >;

C2×C32⋊D6 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes D_6
% in TeX

G:=Group("C2xC3^2:D6");
// GroupNames label

G:=SmallGroup(216,102);
// by ID

G=gap.SmallGroup(216,102);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,201,1444,382,5189,2603]);
// Polycyclic

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

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