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) = C22 |
| direct product | C5×C9×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C9 | C22⋊C4 |
| commutator | C22 | C2×C4×C5×C9 |
| Frattini, cosocle | C23×C3 | C22×C3×C5 |
| socle | C2×C3×C5 | C3×C22⋊C4 |
| prime radical | C2×C3×C5×D4 | C2×C3 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C9 ◁ C23×C5×C9 ◁ G | |
156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14)(15,16)(17,21,22,19)(18,20);(15,17)(16,19)(18,22)(20,21)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 202);