Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#121 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4)(6,8)(9,10,11);(5,6,7,8)(12,13,14);(2,4)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 818);