Finite Group DatabasebetaFinite Group Databasebeta| order | 11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2×C5 |
| direct product | #1044×AGL(1,𝔽11) |
| Fitting, socle, derived | C52×C11 ⋊ C2×C4×C5 |
| Hall | C52×C11⋊C5 ⋊ C2×C4 |
| Sylow | C11 ⋊ C2×C5×#1044 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C11 | C2×C4×C5 |
| prime radical | C5⋊D5×AGL(1,𝔽11) | C2 |
| cosocle | C11×C5⋊D5 | C22×C5 |
| LCS | G ▷ C52×C11 | |
| UCS | C1 | |
216 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 216 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) = 21 |
| transitive degree | μt(G) = 110 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,6,7,9,8,3,4,5,10,11)(12,13,16,14,15)(17,18,19,21,20);(1,2,5,4,8,10,9,11,6,7,3)(13,14,15,16)(18,19,20,21)110 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽11)×D5⋊AGL(1,𝔽5) |
| maximal normal subgroup of | AGL(1,𝔽11)×D5⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,6,7,9,8,3,4,5,10,11)(12,13,16,14,15)(17,18,19,21,20), (1,2,5,4,8,10,9,11,6,7,3)(13,14,15,16)(18,19,20,21));