Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C22×C3 |
| direct product | C5×D5×A42 |
| Fitting, socle | C24×C52 ⋊ C2×C32 |
| derived | C24×C5 ⋊ C2×C32×C5 |
| Hall | C24×C5×D5 ⋊ C32 |
| Sylow | C52 ⋊ C2×A42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D5×A42 |
| commutator, cosocle, hypocenter | C24×C5 | C2×C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C52 | C2×C32 |
| LCS | G ▷ C24×C5 | |
| UCS | C1 ◁ C5 | |
570 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 18 points:
(1,4)(2,3)(5,8,6)(9,10,11,12,13)(15,16)(17,18);(2,4,3)(6,8,7)(14,15,17,18,16)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D52×A42, C5×D5×A4⋊S4 |
| maximal normal subgroup of | D52×A42, C5×D5×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4)(2,3)(5,8,6)(9,10,11,12,13)(15,16)(17,18), (2,4,3)(6,8,7)(14,15,17,18,16));