Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C32 |
| prime radical | C23×C32 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,19);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(20,21,22);(2,6)(5,8)(9,11,12,16)(10,14,13,15)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 823);