Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C22 |
| commutator | C2 | C22×C32×C4 |
| Frattini, cosocle | C22 | C23×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(6,7,8);(3,4,5)(10,12)(13,14,15,16);(9,10,11,12)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 813);