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G = C2×C3⋊S3order 36 = 22·32

Direct product of C2 and C3⋊S3

Aliases: C2×C3⋊S3, C6⋊S3, C32D6, C323C22, (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 >

Character table of C2×C3⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D size 1 1 9 9 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 0 2 -1 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 0 -1 -1 2 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ7 2 -2 0 0 2 -1 -1 -1 1 -2 1 1 orthogonal lifted from D6 ρ8 2 -2 0 0 -1 -1 2 -1 1 1 1 -2 orthogonal lifted from D6 ρ9 2 -2 0 0 -1 -1 -1 2 -2 1 1 1 orthogonal lifted from D6 ρ10 2 2 0 0 -1 -1 -1 2 2 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 0 0 -1 2 -1 -1 1 1 -2 1 orthogonal lifted from D6 ρ12 2 2 0 0 -1 2 -1 -1 -1 -1 2 -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);

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

 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 0 0 0 -1 -1 0 0 0 0 0 1 0 0 1 0
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

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