Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C11 |
| unambiguous semidirect product | D15⋊((C11⋊D11)⋊C3) |
| unambiguous semidirect product | (D15⋊(C11⋊D11))⋊C3 |
| Fitting, socle, derived | C3×C5×C112 ⋊ C22×C3 |
| Hall | C3×C5×C112⋊C3 ⋊ C22 |
| Hall | C3×C112⋊C3 ⋊ C2×D5 |
| Hall | C5×C112 ⋊ C2×C3×D3 |
| Sylow | C112 ⋊ C3×D3×D5 |
| Sylow | C5 ⋊ D3×(C11⋊D11)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C112 | C22×C3 |
| LCS | G ▷ C3×C5×C112 | |
| UCS | C1 | |
168 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3^2,5,11^2|e^a=e^4,f^a=f^10,g^a=g^10,d^b=d^2,e^b=e^4,f^c=f^7g^5,g^c=f^4g^3216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 567);