Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C5 |
| direct product | C23×D3×C5⋊D5 |
| Fitting, socle | C23×C3×C52 ⋊ C22 |
| derived, Hall | C3×C52 ⋊ C25 |
| Sylow | C52 ⋊ C24×D3 |
| Sylow | C3 ⋊ C24×C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×C5⋊D5 |
| commutator, cosocle, hypocenter | C3×C52 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C52 | C22 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C23 | |
5984 conjugacy classes of subgroups; only the 1022 normal subgroups are shown.
This group has 5984 conjugacy classes of subgroups — too many to draw. Only its 1022 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 600 |
A minimal faithful permutation representation — 5 generators on 19 points:
(5,6)(7,8,9)(10,11,13,14,12);(1,2)(3,4)(5,6)(11,12)(13,14)(16,17)(18,19);(8,9)(15,16,18,19,17);(3,4);(1,2)336 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,9)(10,11,13,14,12), (1,2)(3,4)(5,6)(11,12)(13,14)(16,17)(18,19), (8,9)(15,16,18,19,17), (3,4), (1,2));