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## G = C22×S3order 24 = 23·3

### Direct product of C22 and S3

Aliases: C22×S3, C3⋊C23, C6⋊C22, (C2×C6)⋊3C2, SmallGroup(24,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×S3
 Chief series C1 — C3 — S3 — D6 — C22×S3
 Lower central C3 — C22×S3
 Upper central C1 — C22

Generators and relations for C22×S3
G = < a,b,c,d | a2=b2=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of C22×S3

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C size 1 1 1 1 3 3 3 3 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 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 -2 2 -2 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 -2 -2 2 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6

Permutation representations of C22×S3
On 12 points - transitive group 12T10
Generators in S12
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 7)(2 9)(3 8)(4 10)(5 12)(6 11)

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

G:=Group( (1,10)(2,11)(3,12)(4,7)(5,8)(6,9), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,7)(2,9)(3,8)(4,10)(5,12)(6,11) );

G=PermutationGroup([[(1,10),(2,11),(3,12),(4,7),(5,8),(6,9)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,7),(2,9),(3,8),(4,10),(5,12),(6,11)]])

G:=TransitiveGroup(12,10);

Regular action on 24 points - transitive group 24T11
Generators in S24
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 17)(2 16)(3 18)(4 14)(5 13)(6 15)(7 23)(8 22)(9 24)(10 20)(11 19)(12 21)

G:=sub<Sym(24)| (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,17)(2,16)(3,18)(4,14)(5,13)(6,15)(7,23)(8,22)(9,24)(10,20)(11,19)(12,21)>;

G:=Group( (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,17)(2,16)(3,18)(4,14)(5,13)(6,15)(7,23)(8,22)(9,24)(10,20)(11,19)(12,21) );

G=PermutationGroup([[(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,17),(2,16),(3,18),(4,14),(5,13),(6,15),(7,23),(8,22),(9,24),(10,20),(11,19),(12,21)]])

G:=TransitiveGroup(24,11);

C22×S3 is a maximal subgroup of   D6⋊C4
C22×S3 is a maximal quotient of   C4○D12  D42S3  Q83S3

Polynomial with Galois group C22×S3 over ℚ
actionf(x)Disc(f)
12T10x12-4x6+1224·318

Matrix representation of C22×S3 in GL3(ℤ) generated by

 1 0 0 0 -1 0 0 0 -1
,
 -1 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 -1 0 1 -1
,
 -1 0 0 0 0 -1 0 -1 0
G:=sub<GL(3,Integers())| [1,0,0,0,-1,0,0,0,-1],[-1,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,-1,-1],[-1,0,0,0,0,-1,0,-1,0] >;

C22×S3 in GAP, Magma, Sage, TeX

C_2^2\times S_3
% in TeX

G:=Group("C2^2xS3");
// GroupNames label

G:=SmallGroup(24,14);
// by ID

G=gap.SmallGroup(24,14);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,259]);
// Polycyclic

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

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