direct product, metabelian, supersoluble, monomial, A-group, rational, 2-hyperelementary
Aliases: C22×S3, C3⋊C23, C6⋊C22, (C2×C6)⋊3C2, SmallGroup(24,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C22×S3 |
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 |
►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 D4⋊2S3 Q8⋊3S3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T10 | x12-4x6+1 | 224·318 |
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
Export
Subgroup lattice of C22×S3 in TeX
Character table of C22×S3 in TeX