Finite Group DatabasebetaFinite Group Databasebeta| order | 22000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | AGL(1,𝔽11)×D5⋊AGL(1,𝔽5) |
| Fitting, socle, derived | C52×C11 ⋊ C22×C4×C5 |
| Hall | C52×C11⋊C5 ⋊ C22×C4 |
| Sylow | C11 ⋊ C2×C5×D5⋊AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C11 | C22×C4×C5 |
| prime radical | D52×AGL(1,𝔽11) | C2 |
| cosocle | C11×C5⋊D5 | C23×C5 |
| LCS | G ▷ C52×C11 | |
| UCS | C1 | |
552 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 110 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,8,5,11)(3,7,9,6,4)(13,14,15,16)(17,18,19,20);(2,3)(4,6)(5,7)(8,11)(9,10)(12,13,16,14,15);(18,19)(20,21)154 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)2×AGL(1,𝔽11) |
| maximal normal subgroup of | AGL(1,𝔽5)2×AGL(1,𝔽11) |
| holomorph of | D5×C11 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,8,5,11)(3,7,9,6,4)(13,14,15,16)(17,18,19,20), (2,3)(4,6)(5,7)(8,11)(9,10)(12,13,16,14,15), (18,19)(20,21));