Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D15⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | D15⋊(C3⋊D3) |
| unambiguous semidirect product | (C3⋊(C3⋊D3))⋊D5 |
| Fitting, socle, derived, Hall | C33×C5 ⋊ C22 |
| Sylow | C33 ⋊ C2×D5 |
| Sylow | C5 ⋊ D3×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C33×C5 | C22 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
Polycyclic presentation:
2^2,3^3,5|c^a=c^2,d^a=d^2,f^a=f^4,c^b=c^2,d^b=d^2,e^b=e^248 conjugacy classes — view the full character table.
| maximal subgroup of | #4380⋊D3, D3×D5×C3⋊D3, D3×(C3⋊D3)⋊D5, … |
| maximal normal subgroup of | #4380⋊D3, D3×D5×C3⋊D3, D3×(C3⋊D3)⋊D5, … |
| maximal quotient of | #11762087, #11762088, #11762089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 108);