Finite Group DatabasebetaFinite Group Databasebeta| order | 1584 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9×C11 · Schur multiplier (H2) = C22 |
| direct product | C9×C11×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9×C11 | C22 |
| commutator | C2 | C23×C9×C11 |
| Frattini, cosocle | C2×C3 | C23×C3×C11 |
| socle | C2×C3×C11 | C23×C3 |
| prime radical | C3×C11×#41 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9×C11 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 792 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20)(21,22)(23,26)(24,25)(27,28);(22,25)(26,28);(21,23,24,27)(22,26,25,28)990 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1584, 165);