Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×C9×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | D4 |
| commutator | C4 | C23×C9 |
| Frattini, cosocle | C3×C4 | C23×C3 |
| socle, prime radical | C22×C3 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C4×C9 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4,5,6,7,8,9,10)(11,18,17,16,15,14,13,12);(3,11,7,15)(4,12,8,16)(5,13,9,17)(6,14,10,18)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 184);