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G = C2×C32⋊D6  order 216 = 23·33

Direct product of C2 and C32⋊D6

direct product, non-abelian, supersoluble, monomial, rational

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, C3, C3, C22, S3, C6, C6, C23, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, He3, S32, S3×C6, C2×C3⋊S3, C32⋊C6, He3⋊C2, C2×He3, C2×S32, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, C2×C32⋊D6
Quotients: C1, C2, C22, S3, C23, D6, C22×S3, S32, C2×S32, C32⋊D6, C2×C32⋊D6

Character table of C2×C32⋊D6

 class 12A2B2C2D2E2F2G3A3B3C3D6A6B6C6D6E6F6G6H6I6J
 size 119999992661226612181818181818
ρ11111111111111111111111    trivial
ρ21-11-1-11-111111-1-1-1-1111-1-1-1    linear of order 2
ρ3111-1-1-11-1111111111-1-11-1-1    linear of order 2
ρ41-1111-1-1-11111-1-1-1-11-1-1-111    linear of order 2
ρ51-1-11-111-11111-1-1-1-1-11-11-11    linear of order 2
ρ611-1-111-1-111111111-11-1-11-1    linear of order 2
ρ71-1-1-11-1111111-1-1-1-1-1-1111-1    linear of order 2
ρ811-11-1-1-1111111111-1-11-1-11    linear of order 2
ρ92-20-2000222-1-1-2-21100-1001    orthogonal lifted from D6
ρ10220-2000-222-1-122-1-1001001    orthogonal lifted from D6
ρ112-200-22002-12-1-21-210-10010    orthogonal lifted from D6
ρ122-2002-2002-12-1-21-210100-10    orthogonal lifted from D6
ρ132202000222-1-122-1-100-100-1    orthogonal lifted from S3
ρ142-202000-222-1-1-2-21100100-1    orthogonal lifted from D6
ρ15220022002-12-12-12-10-100-10    orthogonal lifted from S3
ρ162200-2-2002-12-12-12-1010010    orthogonal lifted from D6
ρ17440000004-2-214-2-21000000    orthogonal lifted from S32
ρ184-40000004-2-21-422-1000000    orthogonal lifted from C2×S32
ρ196-6-200020-30003000100-100    orthogonal faithful
ρ206-62000-20-30003000-100100    orthogonal faithful
ρ2166200020-3000-3000-100-100    orthogonal lifted from C32⋊D6
ρ2266-2000-20-3000-3000100100    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 14)(8 15)(9 16)(10 17)(11 18)(12 13)

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

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

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

(7 12)(8 11)(9 10)(13 14)(15 18)(16 17)
 
G:=sub<Sym(18)| (1,2)(3,6)(4,5)(7,14)(8,15)(9,16)(10,17)(11,18)(12,13), (1,17,14)(2,10,7)(3,15,16)(4,13,18)(5,12,11)(6,8,9), (1,4,3)(2,5,6)(7,11,9)(8,10,12)(13,15,17)(14,18,16), (3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (7,12)(8,11)(9,10)(13,14)(15,18)(16,17)>;
 
G:=Group( (1,2)(3,6)(4,5)(7,14)(8,15)(9,16)(10,17)(11,18)(12,13), (1,17,14)(2,10,7)(3,15,16)(4,13,18)(5,12,11)(6,8,9), (1,4,3)(2,5,6)(7,11,9)(8,10,12)(13,15,17)(14,18,16), (3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (7,12)(8,11)(9,10)(13,14)(15,18)(16,17) );
 
G=PermutationGroup([[(1,2),(3,6),(4,5),(7,14),(8,15),(9,16),(10,17),(11,18),(12,13)], [(1,17,14),(2,10,7),(3,15,16),(4,13,18),(5,12,11),(6,8,9)], [(1,4,3),(2,5,6),(7,11,9),(8,10,12),(13,15,17),(14,18,16)], [(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(7,12),(8,11),(9,10),(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  C62⋊2D6
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  C62⋊2D6

Matrix representation of C2×C32⋊D6 ►in GL6(ℤ)

-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
000010
000001
100000
010000
001000
000100
,
010000
-1-10000
000100
00-1-100
000001
0000-1-1
,
-100000
110000
00000-1
0000-10
001100
000-100
,
-100000
0-10000
00000-1
000011
001100
00-1000

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
 

Export

Character table of C2×C32⋊D6 in TeX

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