Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C42 |
| direct product | C42⋊C3×#127⋊C3 |
| Fitting, Sylow | C42×#127 ⋊ C32 |
| derived | C44 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C44 | C2×C32 |
| Frattini, socle | C24 | C2×A42 |
| Fitting | C42×#127 | C32 |
| LCS | G ▷ C44 | |
| UCS | C1 | |
1036 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 72 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,9,4,3)(2,7,6,8)(5,11)(10,12)(13,24,19,18,22,14)(15,21,17)(16,23,20);(1,2,5)(3,8,10)(4,6,11)(7,12,9)(13,14,15)(16,21,22)(17,20,24)(18,19,23)80 conjugacy classes — view the full character table.
| maximal subgroup of | #127⋊C32, #72258883 |
| maximal normal subgroup of | #127⋊C32, #72258883 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,9,4,3)(2,7,6,8)(5,11)(10,12)(13,24,19,18,22,14)(15,21,17)(16,23,20), (1,2,5)(3,8,10)(4,6,11)(7,12,9)(13,14,15)(16,21,22)(17,20,24)(18,19,23));