Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of #272⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #272⋊D5 |
| Fitting, derived, Hall | C5×H(3,𝔽3) ⋊ C22 |
| Sylow | H(3,𝔽3) ⋊ C2×D5 |
| Sylow | C5 ⋊ ((C3⋊D3)⋊C3)⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×H(3,𝔽3) | C22 |
| Frattini | C3 | (C3⋊D3)⋊D5 |
| socle | C3×C5 | D32 |
| LCS | G ▷ C5×H(3,𝔽3) | |
| UCS | C1 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 45 |
Polycyclic presentation:
2^2,3^2,5,3|c^a=c^2,e^a=e^4,f^a=f^2,c^b=c^2,d^b=d^2,d^c=df31 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 56);