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) = C22×C3 |
| direct product | C32×C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22 |
| commutator | C2 | C22×C32×C4 |
| Frattini, cosocle | C22 | C23×C32 |
| socle, prime radical | C22×C32 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(7,8,9,10);(4,5,6)(11,12,13,14)(15,18,17,16);(11,15,13,17)(12,16,14,18)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 816);