Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×C4⋊Q8 |
| unambiguous semidirect product | C4⋊(Q8×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C23 |
| commutator | C22 | C23×C9 |
| Frattini, socle, prime radical, cosocle | C22×C3 | C23×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,3,5,7,6,8,9)(11,13)(14,15,17,19)(16,20,21,18);(14,16,17,21)(15,18,19,20);(10,11,12,13)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 178);