Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C5⋊S4)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C33 |
| unambiguous semidirect product | C3⋊(C3⋊(C5⋊S4)) |
| unambiguous semidirect product | C5⋊(C3⋊(C3⋊S4)) |
| unambiguous semidirect product | A4⋊(C3⋊D15) |
| Fitting, socle | C22×C32×C5 ⋊ D3 |
| derived | C32×C5×A4 ⋊ C2 |
| Sylow | C5 ⋊ C3⋊(C3⋊S4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C5×A4 | C2 |
| Fitting, socle | C22×C32×C5 | D3 |
| LCS | G ▷ C32×C5×A4 | |
| UCS | C1 | |
260 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2,3^3,5,2^2|b^a=b^2,c^a=c^2,d^a=d^2,e^a=e^4,f^a=fg,f^b=g,g^b=fg93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 547);