Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C23×C3×D4 (order 192) · Inn(G) = D4 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | D4 |
| commutator | C4 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C3 |
| socle | C2×C3 | C3×C22⋊C4 |
| prime radical | C3×#41 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C2×C4×C9 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,6,4,7,8,5)(9,10,12,11,13,15,14,16,17);(1,2)(3,6)(4,7)(5,8)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 54);