Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×#120 (order 384) · Inn(G) = C22 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×C4⋊C8 |
| unambiguous semidirect product | C4⋊(C8×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C22 |
| commutator | C2 | C2×C8×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C3 |
| socle, prime radical | C22×C3 | C2×C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,4)(5,6,7,9,8,10,11,12)(13,14,16,15,17,19,18,20,21);(1,2,3,4)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 55);