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## G = C4×S3order 24 = 23·3

### Direct product of C4 and S3

Aliases: C4×S3, D6.C2, C122C2, C4Dic3, C2.1D6, Dic32C2, C6.2C22, C31(C2×C4), SmallGroup(24,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C4×S3
 Chief series C1 — C3 — C6 — D6 — C4×S3
 Lower central C3 — C4×S3
 Upper central C1 — C4

Generators and relations for C4×S3
G = < a,b,c | a4=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C4×S3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 12A 12B size 1 1 3 3 2 1 1 3 3 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 i -i i -i -1 -i i linear of order 4 ρ6 1 -1 -1 1 1 i -i -i i -1 -i i linear of order 4 ρ7 1 -1 1 -1 1 -i i -i i -1 i -i linear of order 4 ρ8 1 -1 -1 1 1 -i i i -i -1 i -i linear of order 4 ρ9 2 2 0 0 -1 2 2 0 0 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 -1 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ11 2 -2 0 0 -1 -2i 2i 0 0 1 -i i complex faithful ρ12 2 -2 0 0 -1 2i -2i 0 0 1 i -i complex faithful

Permutation representations of C4×S3
On 12 points - transitive group 12T11
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 8 9)(2 5 10)(3 6 11)(4 7 12)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)

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

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

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

G:=TransitiveGroup(12,11);

Regular action on 24 points - transitive group 24T12
Generators in S24
(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 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)

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

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

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

G:=TransitiveGroup(24,12);

Polynomial with Galois group C4×S3 over ℚ
actionf(x)Disc(f)
12T11x12-24x10+232x8-1152x6+3089x4-4232x2+2312241·172·794

Matrix representation of C4×S3 in GL2(𝔽5) generated by

 3 0 0 3
,
 0 4 1 4
,
 1 4 0 4
G:=sub<GL(2,GF(5))| [3,0,0,3],[0,1,4,4],[1,0,4,4] >;

C4×S3 in GAP, Magma, Sage, TeX

C_4\times S_3
% in TeX

G:=Group("C4xS3");
// GroupNames label

G:=SmallGroup(24,5);
// by ID

G=gap.SmallGroup(24,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,21,259]);
// Polycyclic

G:=Group<a,b,c|a^4=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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