Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C23×#87767 |
| derived, Sylow | C52 ⋊ C24×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | D52 |
| commutator, hypocenter | C52 | C24×C8 |
| Frattini | C4 | C23×D52 |
| Fitting | C23×C4×C52 | C22 |
| socle | C24×C52 | C2×C4 |
| prime radical | C24×C5⋊D5 | C4 |
| cosocle | C4×C52 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23×C4 | |
5314 conjugacy classes of subgroups; only the 1444 normal subgroups are shown.
This group has 5314 conjugacy classes of subgroups — too many to draw. Only its 1444 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)(8,9)(10,11)(12,13)(15,16);(5,6)(7,8,10,11,9);(3,4);(13,15)(14,16)(17,18,19,21,20,22,23,24);(1,2)512 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×#11716761, C22×#13100501, C23×#11716763 |
| maximal normal subgroup of | C23×#11716761, C22×#13100501, C23×#11716763 |
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)(8,9)(10,11)(12,13)(15,16), (5,6)(7,8,10,11,9), (3,4), (13,15)(14,16)(17,18,19,21,20,22,23,24), (1,2));