Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #92030⋊C3 |
| Sylow | C5 ⋊ #1007⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D5×A4 |
| commutator, Fitting, cosocle, hypocenter | C3×C5×Q8 | C22×C3 |
| socle | C2×C3×C5 | C22×A4 |
| LCS | G ▷ C3×C5×Q8 | |
| UCS | C1 ◁ C2 | |
198 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^2,3,2^2,3,5,2|b^2=h,d^2=h,e^2=h,b^a=bh,g^a=g^4,f^b=f^2,d^c=e,e^c=de,e^d=eh75 conjugacy classes — view the full character table.
| maximal subgroup of | #72277397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4912);