Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of C5⋊((C3⋊D3)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊Dic15)⋊C3 |
| derived | C32×C5 ⋊ C3×C4 |
| Hall | C5×H(3,𝔽3) ⋊ C4 |
| Sylow | H(3,𝔽3) ⋊ Dic5 |
| Sylow | C5 ⋊ (C3⋊Dic3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊((C3⋊D3)⋊C3) |
| commutator, hypocenter | C32×C5 | C3×C4 |
| Frattini | C2×C3 | C3×D15 |
| Fitting, prime radical | C2×C5×H(3,𝔽3) | C2 |
| socle | C2×C3×C5 | C3×D3 |
| cosocle | C2×C32×C5 | C2×C3 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2,3,2,3^2,5|a^2=c,d^a=d^2,e^a=e^2,f^a=f^4,d^b=de64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 22);