Finite Group DatabasebetaFinite Group Databasebeta| order | 4800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | A4×#87886 |
| Fitting | C24×C52 ⋊ C22×C3 |
| Hall | C22×#87886 ⋊ C3 |
| Sylow | C52 ⋊ C2×D4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52×A4 |
| commutator, socle, cosocle | C23×C52 | C23×C3 |
| Fitting | C24×C52 | C22×C3 |
| hypercenter | C22 | D52×A4 |
| hypocenter | C22×C52 | C2×C3×D4 |
| LCS | G ▷ C23×C52 ▷ C22×C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1624 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 3 generators on 18 points:
(1,2,4,5,3)(6,7,9,10,8)(12,14)(16,17,18);(2,3)(4,5)(7,8)(9,10)(15,18)(16,17);(7,8)(9,10)(11,12)(13,14)184 conjugacy classes — view the full character table.
| maximal subgroup of | A4×(C5⋊S4)⋊D5, A4×#11717375, #72260259, … |
| maximal normal subgroup of | A4×#11717375, #72260259, #72260358 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,5,3)(6,7,9,10,8)(12,14)(16,17,18), (2,3)(4,5)(7,8)(9,10)(15,18)(16,17), (7,8)(9,10)(11,12)(13,14));