Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of C3⋊D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊Dic45 |
| unambiguous semidirect product | C9⋊Dic15 |
| derived, Hall | C3×C5×C9 ⋊ C4 |
| Sylow | C3×C9 ⋊ Dic5 |
| Sylow | C5 ⋊ C3⋊Dic9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊D45 |
| commutator, hypocenter | C3×C5×C9 | C4 |
| Frattini | C2×C3 | C3⋊D15 |
| Fitting, cosocle | C2×C3×C5×C9 | C2 |
| socle, prime radical | C2×C32×C5 | D3 |
| LCS | G ▷ C3×C5×C9 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 540 |
Polycyclic presentation:
2^2,3^2,5,3|a^2=b,c^3=f^2,c^a=c^2f,d^a=d^2,e^a=e^4,f^a=f^2138 conjugacy classes — view the full character table.
| maximal subgroup of | #11761758, Dic5×C3⋊D9, D5×C3⋊Dic9, … |
| maximal normal subgroup of | #11761758, Dic5×C3⋊D9, D5×C3⋊Dic9, … |
| maximal quotient of | C3⋊(C9⋊#168), C2×C3⋊Dic45, C9⋊Dic45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 26);