Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C11 · Schur multiplier (H2) = C2×C3 |
| direct product | C11×C3⋊D3×AGL(1,𝔽11) |
| unambiguous semidirect product | D11⋊(C5×C11×C3⋊D3) |
| unambiguous semidirect product | (C3×AGL(1,𝔽11))⋊(D3×C11) |
| unambiguous semidirect product | (C3⋊D3×D11)⋊(C5×C11) |
| Fitting, socle, Hall | C32×C112 ⋊ C22×C5 |
| derived | C32×C11 ⋊ C22×C5×C11 |
| Hall | C32×C11×C11⋊C5 ⋊ C22 |
| Hall | C11×C3⋊D3×D11 ⋊ C5 |
| Hall | C11×C11⋊C5 ⋊ C2×C3⋊D3 |
| Sylow | C112 ⋊ C2×C5×C3⋊D3 |
| Sylow | C32 ⋊ C2×C11×AGL(1,𝔽11) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | C3⋊D3×AGL(1,𝔽11) |
| commutator, cosocle, hypocenter | C32×C11 | C22×C5×C11 |
| Frattini | C1 | G |
| Fitting, socle | C32×C112 | C22×C5 |
| LCS | G ▷ C32×C11 | |
| UCS | C1 ◁ C11 | |
252 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
726 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 501);