Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C9 · Schur multiplier (H2) = C2 |
| direct product | C5×C9×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C9 | C22⋊C4 |
| commutator | C22 | C2×C4×C5×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C3×C5 |
| socle | C2×C3×C5 | C3×C22⋊C4 |
| prime radical | C22×C3×C5 | C2×C3×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C9 ◁ C2×C4×C5×C9 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14)(15,16,18,22,19,21,27,29)(17,23,24,30,25,20,26,28);(15,17,19,25)(16,20,21,23)(18,26,27,24)(22,30,29,28)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 204);