Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×C22⋊Q8 |
| 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 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,15);(5,7,8,12)(6,9,10,11)(16,17,18);(2,4)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 819);