Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D3×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D3×D15 |
| unambiguous semidirect product | (C3×D3)⋊(C3×D5) |
| unambiguous semidirect product | (C5×D3)⋊(C3×D3) |
| Fitting, socle, Hall | C33×C5 ⋊ C22 |
| derived | C32×C5 ⋊ C22×C3 |
| Sylow | C33 ⋊ C2×D5 |
| Sylow | C5 ⋊ C3×D32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×D15 |
| commutator, cosocle, hypocenter | C32×C5 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C3 | |
108 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D32×D5, D32×D15, D3×(C3⋊D3)⋊D5, … |
| maximal normal subgroup of | C3×D32×D5, D32×D15, D3×(C3⋊D3)⋊D5, … |
| maximal quotient of | C3×Dic3×D15, C3×D3×Dic15, C3×#67149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 102);