Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×D52×A4 |
| Fitting, socle | C23×C52 ⋊ C22×C3 |
| derived | C22×C52 ⋊ C23×C3 |
| Hall | C23×D52 ⋊ C3 |
| Sylow | C52 ⋊ C23×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D52×A4 |
| commutator, cosocle, hypocenter | C22×C52 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | C22×C3 |
| LCS | G ▷ C22×C52 | |
| UCS | C1 ◁ C2 | |
790 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 790 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(4,5,6)(7,8,10,11,9)(12,13,15,16,14);(3,6)(4,5)(8,9)(10,11)(13,14)(15,16);(13,14)(15,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#87886, C2×A4×D5⋊AGL(1,𝔽5), A4×#87831, … |
| maximal normal subgroup of | A4×#87886, C2×A4×D5⋊AGL(1,𝔽5), A4×#87831, … |
| maximal quotient of | A4×#87886, C22×D52×A4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,5,6)(7,8,10,11,9)(12,13,15,16,14), (3,6)(4,5)(8,9)(10,11)(13,14)(15,16), (13,14)(15,16));