Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210×C5 |
| direct product | C23×#87862 |
| derived, Sylow | C52 ⋊ C24×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | #1044 |
| commutator, hypocenter | C52 | C24×C8 |
| Frattini | C2 | C24×#1044 |
| Fitting | C23×C4×C52 | C4 |
| socle | C24×C52 | C2×C4 |
| prime radical | C24×C5⋊D5 | C4 |
| cosocle | #1039 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23×C4 | |
5314 conjugacy classes of subgroups; only the 1342 normal subgroups are shown.
This group has 5314 conjugacy classes of subgroups — too many to draw. Only its 1342 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) = 320 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8,11,9,10)(12,13,14,16,15)(17,19,21,20)(18,22,24,23);(5,6);(3,4);(8,9,10,11)(13,14,15,16)(17,18,20,23,21,24,19,22);(1,2)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×#11716761, C23×#11716763, C22×#13104814 |
| maximal normal subgroup of | C23×#11716761, C23×#11716763, C22×#13104814 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,6)(7,8,11,9,10)(12,13,14,16,15)(17,19,21,20)(18,22,24,23), (5,6), (3,4), (8,9,10,11)(13,14,15,16)(17,18,20,23,21,24,19,22), (1,2));