Finite Group DatabasebetaFinite Group Databasebeta| order | 7920 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22×C3 |
| direct product | PSU(3,𝔽2)×AGL(1,𝔽11) |
| Fitting, socle, Hall | C32×C11 ⋊ C2×C5×Q8 |
| Hall | D11×PSU(3,𝔽2) ⋊ C5 |
| Hall | C32×C11⋊C5 ⋊ C2×Q8 |
| Hall | C11⋊C5 ⋊ C2×PSU(3,𝔽2) |
| Sylow | C11 ⋊ C2×C5×PSU(3,𝔽2) |
| Sylow | C32 ⋊ Q8×AGL(1,𝔽11) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C11×C3⋊D3 | C23×C5 |
| Fitting, socle, hypocenter | C32×C11 | C2×C5×Q8 |
| prime radical | C3⋊D3×AGL(1,𝔽11) | C22 |
| LCS | G ▷ C11×C3⋊D3 ▷ C32×C11 | |
| UCS | C1 | |
172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 99 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,4,5,3)(2,6,8,7)(10,11,16,12,18)(13,20,19,14,17);(2,4,6,7)(3,5,8,9);(11,12)(13,15)(14,17)(16,20)(18,19)66 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽11)×PSU(3,𝔽2)⋊C3, AGL(1,𝔽11)×AΓL(1,𝔽9) |
| maximal normal subgroup of | AGL(1,𝔽11)×PSU(3,𝔽2)⋊C3, AGL(1,𝔽11)×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,5,3)(2,6,8,7)(10,11,16,12,18)(13,20,19,14,17), (2,4,6,7)(3,5,8,9), (11,12)(13,15)(14,17)(16,20)(18,19));