Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of #1088 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#1088 |
| Fitting, Hall | C5×H(3,𝔽3) ⋊ C4 |
| derived, Sylow | H(3,𝔽3) ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3×C5 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C4×C5 |
| Frattini | C3 | C5×#157 |
| Fitting | C5×H(3,𝔽3) | C4 |
| prime radical | C5×#272 | C2 |
| cosocle | #272 | C2×C5 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C5 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#6249, #11761920, D5×#1088, … |
| maximal normal subgroup of | C5×#6249, #11761920, D5×#1088, … |
| maximal quotient of | C5×#6188, C2×C5×#1088, C3×C5×#1088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 41);