Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×#2818⋊C3 |
| Fitting, Sylow | C22×#2818 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #560⋊C3 |
| commutator, cosocle | C2×C42 | C24×C3 |
| Frattini | C23 | C24×A4 |
| Fitting | C22×#2818 | C3 |
| socle | C25 | C22×A4 |
| hypocenter | C42 | C22×C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C23 | |
5320 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 5320 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,4)(10,16,22)(11,17,21)(12,13,20)(14,19,24)(15,18,23);(1,2)(3,4)(5,6)(7,8)(9,24)(10,18)(11,21)(12,22)(13,23)(14,16)(15,19)(17,20);(1,2)(3,4)(6,8)(10,14)(11,18)(15,21)(16,19)(17,23)(22,24);(1,2)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213963, C2×#72213984 |
| maximal normal subgroup of | C2×#72213963, C2×#72213984 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(10,16,22)(11,17,21)(12,13,20)(14,19,24)(15,18,23), (1,2)(3,4)(5,6)(7,8)(9,24)(10,18)(11,21)(12,22)(13,23)(14,16)(15,19)(17,20), (1,2)(3,4)(6,8)(10,14)(11,18)(15,21)(16,19)(17,23)(22,24), (1,2));