Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C3⋊(C5⋊(#126⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊(C5⋊(#126⋊C3)) |
| unambiguous semidirect product | C5⋊(C3⋊(#126⋊C3)) |
| unambiguous semidirect product | #91765⋊C3 |
| Fitting, derived | C3×C42×C5 ⋊ C2×C3 |
| Sylow | C5 ⋊ C3⋊(#126⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C42×C5 | C2×C3 |
| Frattini | C22 | A4×D15 |
| socle | C22×C3×C5 | C2×A4 |
| LCS | G ▷ C3×C42×C5 | |
| UCS | C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2,3,2^2,3,5,2^2|c^2=g,d^2=h,c^a=ch,d^a=dgh,e^a=e^2,f^a=f^4,c^b=d,d^b=cdgh,g^b=h,h^b=gh66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2553);