Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D15⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D15⋊D9 |
| unambiguous semidirect product | (C3⋊D9)⋊D5 |
| unambiguous semidirect product | D45⋊D3 |
| Fitting, derived, Hall | C3×C5×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ C2×D5 |
| Sylow | C5 ⋊ D3×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C5×C9 | C22 |
| Frattini | C3 | (C3⋊D3)⋊D5 |
| socle | C32×C5 | C2×D3 |
| LCS | G ▷ C3×C5×C9 | |
| UCS | C1 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
Polycyclic presentation:
2^2,3^2,5,3|d^3=f^2,c^a=c^2,e^a=e^4,c^b=c^2,d^b=d^2f,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×D9, C2×D15⋊D9, #4169⋊D5, … |
| maximal normal subgroup of | D3×D5×D9, C2×D15⋊D9, C3×D15⋊D9, … |
| maximal quotient of | #11761805, #11761806, (C3⋊D9)⋊Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 55);