direct product, abelian, monomial, 2-elementary
Aliases: C22xC6, SmallGroup(24,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22xC6 |
| C1 — C22xC6 |
| C1 — C22xC6 |
Generators and relations for C22xC6
G = < a,b,c | a2=b2=c6=1, ab=ba, ac=ca, bc=cb >
Quotients: C1, C2, C3, C22, C6, C23, C2xC6, C22xC6
Character table of C22xC6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ12 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ13 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ16 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ17 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ18 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ19 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ20 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ21 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ22 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ23 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ24 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | linear of order 6 |
►Regular action on 24 points - transitive group 24T3
Generators in S24
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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)
G:=sub<Sym(24)| (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
G:=Group( (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
G=PermutationGroup([[(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])
G:=TransitiveGroup(24,3);
C22xC6 is a maximal subgroup of
C6.D4
Matrix representation of C22xC6 ►in GL3(F7) generated by
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 3 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 5 |
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,6],[6,0,0,0,1,0,0,0,1],[3,0,0,0,2,0,0,0,5] >;
C22xC6 in GAP, Magma, Sage, TeX
C_2^2\times C_6
% in TeX
G:=Group("C2^2xC6"); // GroupNames label
G:=SmallGroup(24,15);
// by ID
G=gap.SmallGroup(24,15);
# by ID
G:=PCGroup([4,-2,-2,-2,-3]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of C22xC6 in TeX
Character table of C22xC6 in TeX