Finite Group DatabasebetaFinite Group Databasebeta| order | 1584 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C11 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×C11×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4×C11 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C32×C11 |
| socle | C2×C32×C11 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4×C11 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 792 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,23)(21,22)(24,25);(4,5,6)(19,22)(23,25);(18,20,21,24)(19,23,22,25)990 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1584, 651);