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) = C2×C3×C4 |
| direct product | C32×C5×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4×C5 | C22 |
| commutator | C2 | C22×C32×C4×C5 |
| Frattini, cosocle | C22 | C23×C32×C5 |
| socle | C22×C32×C5 | C23 |
| prime radical | C23×C32×C5 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4×C5 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(7,8,9,10,11)(12,13,14,15)(16,17)(18,21)(19,20)(22,23);(4,5,6)(17,20)(21,23);(16,18,19,22)(17,21,20,23)900 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4543);