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G = C2×C3⋊S3  order 36 = 22·32

Direct product of C2 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C2×C3⋊S3, C6⋊S3, C3⋊2D6, C32⋊3C22, (C3×C6)⋊2C2, SmallGroup(36,13)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×C3⋊S3
 G = < a,b,c,d | a2=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

C1
C2
9C2
9C2
C3
C3
C3
C3
9C22
C6
C6
C6
C6
3S3
3S3
3S3
3S3
3S3
3S3
3S3
3S3
C32
3D6
3D6
3D6
3D6
C3⋊S3
C3×C6
C3⋊S3
C2×C3⋊S3

Character table of C2×C3⋊S3

 class 12A2B2C3A3B3C3D6A6B6C6D
 size 119922222222
ρ1111111111111    trivial
ρ211-1-111111111    linear of order 2
ρ31-11-11111-1-1-1-1    linear of order 2
ρ41-1-111111-1-1-1-1    linear of order 2
ρ522002-1-1-1-12-1-1    orthogonal lifted from S3
ρ62200-1-12-1-1-1-12    orthogonal lifted from S3
ρ72-2002-1-1-11-211    orthogonal lifted from D6
ρ82-200-1-12-1111-2    orthogonal lifted from D6
ρ92-200-1-1-12-2111    orthogonal lifted from D6
ρ102200-1-1-122-1-1-1    orthogonal lifted from S3
ρ112-200-12-1-111-21    orthogonal lifted from D6
ρ122200-12-1-1-1-12-1    orthogonal lifted from S3

Permutation representations of C2×C3⋊S3
►On 18 points - transitive group 18T12
Generators in S18
(1 11)(2 12)(3 10)(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 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)

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

C2×C3⋊S3 is a maximal subgroup of   C6.D6  C3⋊D12  C12⋊S3  C32⋊7D4  C2×S32  C6.6S4
C2×C3⋊S3 is a maximal quotient of   C32⋊4Q8  C12⋊S3  C32⋊7D4

Matrix representation of C2×C3⋊S3 ►in GL4(ℤ) generated by

1000
0100
00-10
000-1
,
1000
0100
00-11
00-10
,
-1-100
1000
000-1
001-1
,
1000
-1-100
0001
0010
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,-1,-1,0,0,1,0],[-1,1,0,0,-1,0,0,0,0,0,0,1,0,0,-1,-1],[1,-1,0,0,0,-1,0,0,0,0,0,1,0,0,1,0] >;
 

C2×C3⋊S3 in GAP, Magma, Sage, TeX 

C_2\times C_3\rtimes S_3
 
% in TeX
 
G:=Group("C2xC3:S3");
 
// GroupNames label
 
G:=SmallGroup(36,13);
 
// by ID
 
G=gap.SmallGroup(36,13);
 
# by ID
 
G:=PCGroup([4,-2,-2,-3,-3,98,387]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C3⋊S3 in TeX
Character table of C2×C3⋊S3 in TeX

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