Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#91185 |
| derived | C3×C5×SL(2,𝔽3) ⋊ C4 |
| Sylow | C5 ⋊ C3⋊GU(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C3⋊(C5⋊S4) |
| commutator, hypocenter | C3×C5×SL(2,𝔽3) | C4 |
| Fitting | C3×C5×#41 | D3 |
| socle | C2×C3×C5 | SL(2,ℤ/4) |
| prime radical, cosocle | C3×C5×#41⋊C3 | C2 |
| LCS | G ▷ C3×C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^2,3^2,5,2^3|a^2=b,b^2=h,f^2=h,g^2=h,c^a=c^2,d^a=d^2,e^a=e^4,f^a=fg,g^a=gh,f^c=g,g^c=fgh,g^f=gh114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2536);