Finite Group DatabasebetaFinite Group Databasebeta| order | 47520 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽11)×AGL(2,𝔽3) |
| Fitting, socle | C32×C11 ⋊ C2×C5×GL(2,𝔽3) |
| derived | C11×PSU(3,𝔽2)⋊C3 ⋊ C22×C5 |
| Hall | D11×AGL(2,𝔽3) ⋊ C5 |
| Hall | C11⋊C5 ⋊ C2×AGL(2,𝔽3) |
| Sylow | C11 ⋊ C2×C5×AGL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C11×PSU(3,𝔽2)⋊C3 | C22×C5 |
| Fitting, socle | C32×C11 | C2×C5×GL(2,𝔽3) |
| LCS | G ▷ C11×PSU(3,𝔽2)⋊C3 | |
| UCS | C1 | |
576 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 34 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 — 2 generators on 20 points:
(1,7,4)(2,6,3)(5,9,8)(11,15,16,14,19,12,13,20,17,18);(2,4)(3,6)(7,8)(10,11,14,19,13,16,20,15,18,17,12)121 conjugacy classes — view the full character table.
| automorphism group of | C3⋊D33 |
| holomorph of | C32×C11 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,4)(2,6,3)(5,9,8)(11,15,16,14,19,12,13,20,17,18), (2,4)(3,6)(7,8)(10,11,14,19,13,16,20,15,18,17,12));