Finite Group DatabasebetaFinite Group Databasebeta| order | 1584 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C11 · Schur multiplier (H2) = C3 |
| direct product | C32×C11×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C32×C11 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C32×C11 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C11 ◁ C32×C4×C11 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1584 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3)(7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25)(26,33,32,31,30,29,28,27);(4,5,6)(18,26,22,30)(19,27,23,31)(20,28,24,32)(21,29,25,33)693 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1584, 363);