Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×D5×#997 |
| Fitting | C3×C5×H(5,𝔽2) ⋊ C2×C3 |
| derived | C5×H(5,𝔽2) ⋊ C2×C32 |
| Hall | D5×H(5,𝔽2) ⋊ C32 |
| Sylow | C5 ⋊ C2×C3×#997 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5×C22⋊A4 |
| commutator, cosocle, hypocenter | C5×H(5,𝔽2) | C2×C32 |
| Frattini | C2 | C3×D5×C22⋊A4 |
| Fitting | C3×C5×H(5,𝔽2) | C2×C3 |
| socle | C2×C3×C5 | C2×C22⋊A4 |
| LCS | G ▷ C5×H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
876 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 120 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(3,8)(10,11)(12,13)(14,15,16);(2,5,3)(6,7,8)(9,10,12,13,11);(1,7,4,5)(2,3,6,8)132 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽5)×#997, D3×D5×#997, ΓL(1,𝔽16)×#997, … |
| maximal normal subgroup of | C3×AGL(1,𝔽5)×#997, D3×D5×#997, ΓL(1,𝔽16)×#997, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,6)(3,8)(10,11)(12,13)(14,15,16), (2,5,3)(6,7,8)(9,10,12,13,11), (1,7,4,5)(2,3,6,8));