Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4×C5 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C32×C5×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32×C5 | C22 |
| commutator | C2 | C22×C32×C4×C5 |
| Frattini, cosocle | C22 | C23×C32×C5 |
| prime radical | C24×C32×C5 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32×C5 ◁ G | |
792 conjugacy classes of subgroups; only the 456 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 456 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(11,12,13)(17,18,19,20,21);(3,4,6,8)(5,7,9,10)(14,15,16);(4,7)(8,10)900 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4541);