Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×ΓL(1,𝔽16) |
| unambiguous semidirect product | C3⋊(C32×AGL(1,𝔽5)) |
| Fitting, socle, Hall | C33×C5 ⋊ C4 |
| derived | C3×C5 ⋊ C32×C4 |
| Sylow | C33 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ C32×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C33×C5 | C4 |
| prime radical | C33×D5 | C2 |
| cosocle | C3×D5 | C2×C32 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C32 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 90);