Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C32 |
| direct product | C22×C32×PSU(3,𝔽2) |
| Fitting, socle | C22×C34 ⋊ Q8 |
| Sylow | C34 ⋊ C22×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | PSU(3,𝔽2) |
| commutator, cosocle | C3⋊D3 | C24×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | Q8 |
| prime radical | C22×C32×C3⋊D3 | C22 |
| hypocenter | C32 | C22×C32×Q8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
1142 conjugacy classes of subgroups; only the 498 normal subgroups are shown.
This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 498 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 19 points:
(4,5,6)(7,15,8,9)(10,14,13,12);(1,2,3)(8,9,11,14)(10,15,12,13);(18,19);(16,17)216 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C32×PSU(3,𝔽2)⋊C3, C22×C3⋊D3×PSU(3,𝔽2), C22×C32×AΓL(1,𝔽9), … |
| maximal normal subgroup of | C22×C32×PSU(3,𝔽2)⋊C3, C22×C3⋊D3×PSU(3,𝔽2), C22×C32×AΓL(1,𝔽9), … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(7,15,8,9)(10,14,13,12), (1,2,3)(8,9,11,14)(10,15,12,13), (18,19), (16,17));