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×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | C22 |
| commutator | C2 | C22×C32×C4 |
| Frattini, cosocle | C4 | C23×C32 |
| socle | C2×C32 | C22×C4 |
| prime radical | C32×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ G | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,19);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(20,21,22);(3,9)(6,12)(10,15)(13,16)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 828);