Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×#41 (order 192) · Inn(G) = C22 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C9 | C22 |
| commutator | C2 | C2×C8×C9 |
| Frattini, cosocle | C3×C8 | C22×C3 |
| socle | C2×C3 | C2×C3×C8 |
| prime radical | C22×C3 | C3×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C9 ◁ G | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(11,19)(13,21)(15,23)(17,25)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 60);