direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×S3, D6.C2, C12⋊2C2, C4○Dic3, C2.1D6, Dic3⋊2C2, C6.2C22, C3⋊1(C2×C4), SmallGroup(24,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C4×S3 |
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 |
►On 12 points - transitive group 12T11
(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
(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);
C4×S3 is a maximal subgroup of
C4.6S4 A5⋊C4 C4.A5
C2p.D6: C8⋊S3 C4○D12 D4⋊2S3 Q8⋊3S3 C6.D6 D30.C2 D21⋊C4 D33⋊C4 ...
C4×S3 is a maximal quotient of
C8⋊S3
C6.D2p: Dic3⋊C4 D6⋊C4 C6.D6 D30.C2 D21⋊C4 D33⋊C4 D78.C2 D51⋊2C4 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 12T11 | x12-24x10+232x8-1152x6+3089x4-4232x2+2312 | 241·172·794 |
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
Export
Subgroup lattice of C4×S3 in TeX
Character table of C4×S3 in TeX