Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C3 |
| direct product | C32×C5×Dic3 |
| unambiguous semidirect product | C3⋊(C32×C4×C5) |
| derived | C3 ⋊ C32×C4×C5 |
| Hall | C33×C5 ⋊ C4 |
| Sylow | C33 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C5 | D3 |
| commutator, hypocenter | C3 | C32×C4×C5 |
| Frattini | C2 | C32×C5×D3 |
| Fitting, socle, prime radical | C2×C33×C5 | C2 |
| cosocle | C2×C3 | C2×C32×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C32×C5 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×D5×Dic3, C32×#1200, C32×#1202, … |
| maximal normal subgroup of | C32×D5×Dic3, C32×#1200, C32×#1202, … |
| maximal quotient of | C32×C5×C3⋊C8, C2×C32×C5×Dic3, C32×C5×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 81);