Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | ((C11⋊C5)⋊D11)⋊(C3⋊D3) |
| Fitting, socle, derived, Hall | C32×C112 ⋊ C22×C5 |
| Hall | C32×C11×C11⋊C5 ⋊ C22 |
| Hall | #17983775 ⋊ C5 |
| Hall | C11×C11⋊C5 ⋊ C2×C3⋊D3 |
| Sylow | C112 ⋊ C2×C5×C3⋊D3 |
| Sylow | C32 ⋊ D11×AGL(1,𝔽11) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C112 | C22×C5 |
| LCS | G ▷ C32×C112 | |
| UCS | C1 | |
252 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,5,3^2,11^2|d^a=d^2,e^a=e^2,f^a=f^10,f^b=f^10,g^b=g^10,g^c=g^3261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 505);