Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C2×C9×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C22 |
| commutator | C2 | C22×C4×C9 |
| Frattini, cosocle | C22×C3 | C23×C3 |
| socle | C23×C3 | C22×C3 |
| prime radical | C24×C3 | C2×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(11,12,14,13,15,17,16,18,19);(4,6)(7,8,9,10);(3,4)(5,6)(7,8)(9,10)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 165);