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 | C32×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22 |
| commutator | C2 | C22×C32×C4 |
| Frattini, cosocle | C22 | C23×C32 |
| socle | C22×C32 | C23 |
| prime radical | C2×C32×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ 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)(7,8,9,10);(4,5,6)(11,12,13,14);(12,14)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 815);