Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×D32 |
| unambiguous semidirect product | (C3×D3)⋊(C2×C3×C5) |
| Fitting, socle, Hall | C33×C5 ⋊ C22 |
| derived | C32 ⋊ C22×C3×C5 |
| Sylow | C33 ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | D32 |
| commutator, cosocle, hypocenter | C32 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C5 | |
108 conjugacy classes of subgroups.
| transitive degree | μt(G) = 60 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×GO+(4,𝔽2), C3×D32⋊D5, C5×D32⋊D3, … |
| maximal normal subgroup of | C3×C5×GO+(4,𝔽2), C3×D32⋊D5, C5×D32⋊D3, … |
| maximal quotient of | C3×C5×D3×Dic3, C3×C5×#624, C3×C5×#625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 101);