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G = C2×C32⋊D6order 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

C1C3He3 — C2×C32⋊D6
C1C3C32He3C32⋊C6C32⋊D6 — C2×C32⋊D6
He3 — C2×C32⋊D6
C1C2

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 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 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)

-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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