Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×#6038 |
| Hall | C22×#6038 ⋊ C3 |
| Sylow | C52 ⋊ D4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D52×A4 |
| commutator, Fitting, socle, cosocle | C23×C52 | C22×C3 |
| hypocenter | C22×C52 | C3×D4 |
| LCS | G ▷ C23×C52 ▷ C22×C52 | |
| UCS | C1 ◁ C2 | |
342 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 60 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5)(6,7)(8,9)(12,14)(16,17,18);(1,2,4,5,3)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)116 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#87886, #72260214, #72260232 |
| maximal normal subgroup of | A4×#87886, #72260214, #72260232 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,7)(8,9)(12,14)(16,17,18), (1,2,4,5,3)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17));