Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×C3×#442 (order 384) · Inn(G) = D8 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | D8 |
| commutator | C8 | C22×C9 |
| Frattini, cosocle | C3×C8 | C22×C3 |
| socle | C2×C3 | C3×D8 |
| prime radical | C3×D8 | C2×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C4×C9 ◁ 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);(11,17)(12,24)(13,15)(14,22)(16,20)(19,25)(21,23)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 62);