Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×ΓL(1,𝔽16)⋊AGL(1,𝔽5) |
| Fitting, socle, Hall | C32×C52 ⋊ C42 |
| derived | C3×C52 ⋊ C3×C42 |
| Sylow | C52 ⋊ C3×C4×Dic3 |
| Sylow | C32 ⋊ AGL(1,𝔽5)2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽16)⋊AGL(1,𝔽5) |
| commutator, hypocenter | C3×C52 | C3×C42 |
| Frattini | C1 | G |
| Fitting, socle | C32×C52 | C42 |
| prime radical | C32×D52 | C22 |
| cosocle | C3×D52 | C22×C3 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C3 | |
290 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 16 points:
(1,2,3)(4,5,6)(8,9,10,11)(12,13,14,15);(5,6)(7,8,11,9,10)(13,15,14,16)117 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,6)(8,9,10,11)(12,13,14,15), (5,6)(7,8,11,9,10)(13,15,14,16));