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) = C22×C3 |
| direct product | C32×C5×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C5×C8 | C22 |
| commutator | C2 | C22×C32×C4×C5 |
| Frattini, cosocle | C4 | C23×C32×C5 |
| socle | C2×C32×C5 | C22×C4 |
| prime radical | C32×C5×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C5×C8 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(7,8,9,10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27);(4,5,6)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(14,20)(17,23)(21,26)(24,27)900 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4557);