Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C24 |
| direct product | C22×C5×D5×A4 |
| Fitting, socle | C24×C52 ⋊ C2×C3 |
| derived | C22×C5 ⋊ C23×C3×C5 |
| Hall | C24×C5×D5 ⋊ C3 |
| Sylow | C52 ⋊ C23×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | D5×A4 |
| commutator, cosocle, hypocenter | C22×C5 | C23×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C52 | C2×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C22×C5 | |
790 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 790 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 18 points:
(3,4)(6,8,7)(9,10,11,12,13)(14,15,17,18,16);(1,2)(5,8)(6,7);(15,16)(17,18)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×D5×A42, C22×D52×A4, C5×D5×GL(2,ℤ/4), … |
| maximal normal subgroup of | C5×D5×A42, C22×D52×A4, C5×D5×GL(2,ℤ/4), … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(6,8,7)(9,10,11,12,13)(14,15,17,18,16), (1,2)(5,8)(6,7), (15,16)(17,18));