Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D5×D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D5×D27 |
| unambiguous semidirect product | (C5×D27)⋊C2 |
| unambiguous semidirect product | (D5×C27)⋊C2 |
| unambiguous semidirect product | D135⋊C2 |
| Fitting, derived, Hall | C5×C27 ⋊ C22 |
| Sylow | C27 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×D27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×C27 | C22 |
| Frattini | C9 | D3×D5 |
| socle | C3×C5 | C2×D9 |
| LCS | G ▷ C5×C27 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×D27, C2×D5×D27, D5×D81, … |
| maximal normal subgroup of | AGL(1,𝔽5)×D27, C2×D5×D27, C3×D5×D27, … |
| maximal quotient of | #11761661, Dic5×D27, #11761663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 7);