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×C4 |
| direct product | C32×#124 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C32 |
| prime radical | C2×C32×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,15);(1,2,4,3)(5,7,8,11)(6,9,10,12)(16,17,18);(1,3,4,2)(6,10)(9,12)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 821);