Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C3 |
| direct product | C3×C5×#157 |
| Fitting, socle, Hall | C33×C5 ⋊ C4 |
| derived | C32 ⋊ C3×C4×C5 |
| Sylow | C33 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | #157 |
| commutator, hypocenter | C32 | C3×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C4 |
| prime radical | C3×C5×C3⋊D3 | C2 |
| cosocle | C3⋊D3 | C2×C3×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C5 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 60 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×AGL(1,𝔽9), C3×#67194, C5×C3⋊AGL(1,𝔽9), … |
| maximal normal subgroup of | C3×C5×AGL(1,𝔽9), C3×#67194, C5×C3⋊AGL(1,𝔽9), … |
| maximal quotient of | C3×C5×#622, C2×C3×C5×#157, C5×C9×#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 93);