Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = C2×C3×B(4,𝔽2) (order 384) · Inn(G) = C22⋊C4 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C22⋊C4 |
| commutator | C22 | C2×C4×C9 |
| Frattini, cosocle | C23×C3 | C22×C3 |
| socle | C2×C3 | C3×C22⋊C4 |
| prime radical | C2×C3×D4 | C2×C3 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C23×C9 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,3,5,7,6,8,9)(10,11)(12,16,17,14)(13,15);(10,12)(11,14)(13,17)(15,16)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 49);