Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#20535 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,9)(10,13)(11,12)(14,15)(16,17,20,23,19,21,18,22);(4,5,6,7)(9,12)(10,14)(17,21)(22,23);(8,10,11,14)(9,13,12,15)(16,18,19,20)(17,22,21,23);(9,12)(13,15)(17,21)(22,23)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13523);