direct product, cyclic, abelian, monomial
Aliases: C24, also denoted Z24, SmallGroup(24,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C24 |
| C1 — C24 |
| C1 — C24 |
Generators and relations for C24
G = < a | a24=1 >
Character table of C24
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| 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 | i | -i | -1 | -1 | ζ8 | ζ85 | ζ83 | ζ87 | i | -i | i | -i | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ8 | linear of order 8 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -1 | -i | i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ5 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ83 | ζ87 | ζ8 | ζ85 | -i | i | -i | i | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ83 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ85 | ζ8 | ζ87 | ζ83 | i | -i | i | -i | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ85 | linear of order 8 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | -1 | i | -i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ87 | ζ83 | ζ85 | ζ8 | -i | i | -i | i | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ87 | linear of order 8 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | ζ8 | ζ85 | ζ83 | ζ87 | ζ82ζ3 | ζ86ζ3 | ζ82ζ32 | ζ86ζ32 | ζ87ζ32 | ζ85ζ3 | ζ83ζ3 | ζ87ζ3 | ζ8ζ32 | ζ85ζ32 | ζ83ζ32 | ζ8ζ3 | linear of order 24 faithful |
| ρ11 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | i | i | -i | -i | ζ65 | ζ65 | ζ6 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | ζ83 | ζ87 | ζ8 | ζ85 | ζ86ζ3 | ζ82ζ3 | ζ86ζ32 | ζ82ζ32 | ζ85ζ32 | ζ87ζ3 | ζ8ζ3 | ζ85ζ3 | ζ83ζ32 | ζ87ζ32 | ζ8ζ32 | ζ83ζ3 | linear of order 24 faithful |
| ρ13 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ14 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | ζ85 | ζ8 | ζ87 | ζ83 | ζ82ζ3 | ζ86ζ3 | ζ82ζ32 | ζ86ζ32 | ζ83ζ32 | ζ8ζ3 | ζ87ζ3 | ζ83ζ3 | ζ85ζ32 | ζ8ζ32 | ζ87ζ32 | ζ85ζ3 | linear of order 24 faithful |
| ρ15 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | -i | -i | i | i | ζ65 | ζ65 | ζ6 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | linear of order 12 |
| ρ16 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | ζ87 | ζ83 | ζ85 | ζ8 | ζ86ζ3 | ζ82ζ3 | ζ86ζ32 | ζ82ζ32 | ζ8ζ32 | ζ83ζ3 | ζ85ζ3 | ζ8ζ3 | ζ87ζ32 | ζ83ζ32 | ζ85ζ32 | ζ87ζ3 | linear of order 24 faithful |
| ρ17 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ18 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | ζ8 | ζ85 | ζ83 | ζ87 | ζ82ζ32 | ζ86ζ32 | ζ82ζ3 | ζ86ζ3 | ζ87ζ3 | ζ85ζ32 | ζ83ζ32 | ζ87ζ32 | ζ8ζ3 | ζ85ζ3 | ζ83ζ3 | ζ8ζ32 | linear of order 24 faithful |
| ρ19 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | i | i | -i | -i | ζ6 | ζ6 | ζ65 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | linear of order 12 |
| ρ20 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | ζ83 | ζ87 | ζ8 | ζ85 | ζ86ζ32 | ζ82ζ32 | ζ86ζ3 | ζ82ζ3 | ζ85ζ3 | ζ87ζ32 | ζ8ζ32 | ζ85ζ32 | ζ83ζ3 | ζ87ζ3 | ζ8ζ3 | ζ83ζ32 | linear of order 24 faithful |
| ρ21 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ22 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | ζ85 | ζ8 | ζ87 | ζ83 | ζ82ζ32 | ζ86ζ32 | ζ82ζ3 | ζ86ζ3 | ζ83ζ3 | ζ8ζ32 | ζ87ζ32 | ζ83ζ32 | ζ85ζ3 | ζ8ζ3 | ζ87ζ3 | ζ85ζ32 | linear of order 24 faithful |
| ρ23 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | -i | -i | i | i | ζ6 | ζ6 | ζ65 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | linear of order 12 |
| ρ24 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | ζ87 | ζ83 | ζ85 | ζ8 | ζ86ζ32 | ζ82ζ32 | ζ86ζ3 | ζ82ζ3 | ζ8ζ3 | ζ83ζ32 | ζ85ζ32 | ζ8ζ32 | ζ87ζ3 | ζ83ζ3 | ζ85ζ3 | ζ87ζ32 | linear of order 24 faithful |
►Regular action on 24 points - transitive group 24T1
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)
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)>;
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) );
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)]])
G:=TransitiveGroup(24,1);
C24 is a maximal subgroup of
C3⋊C16 C8⋊S3 C24⋊C2 D24 Dic12 C8.A4 C7⋊C24 He3⋊2C8 C13⋊2C24 C13⋊C24 C19⋊C24
C24 is a maximal quotient of
C7⋊C24 C13⋊2C24 C13⋊C24 C19⋊C24
Matrix representation of C24 ►in GL1(𝔽73) generated by
| 21 |
G:=sub<GL(1,GF(73))| [21] >;
C24 in GAP, Magma, Sage, TeX
C_{24} % in TeX
G:=Group("C24"); // GroupNames label
G:=SmallGroup(24,2);
// by ID
G=gap.SmallGroup(24,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-2,24,34]);
// Polycyclic
G:=Group<a|a^24=1>;
// generators/relations
Export
Subgroup lattice of C24 in TeX
Character table of C24 in TeX