Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #11925896⋊D9 |
| Fitting, derived, Hall | C9×C112 ⋊ C22×C5 |
| Hall | C9×#10597245 ⋊ C22 |
| Hall | D11×D99 ⋊ C5 |
| Hall | #10597245 ⋊ C2×D9 |
| Sylow | C112 ⋊ C2×C5×D9 |
| Sylow | C9 ⋊ #11925896⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C9×C112 | C22×C5 |
| Frattini | C3 | #11925896⋊D3 |
| socle | C3×C112 | C2×C5×D3 |
| LCS | G ▷ C9×C112 | |
| UCS | C1 | |
126 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,5,3,11^2,3|d^3=g^2,d^a=d^2g,f^a=f^10,g^a=g^2,e^b=e^10,e^c=e^4,f^c=f^3129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 189);