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×C42 |
| direct product | C3×#13818 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,5,7,6)(8,9,11,10)(12,13,15,14)(17,20)(21,23);(5,6)(12,13,15,14);(8,9,11,10)(12,13,15,14)(16,17)(18,21)(19,20)(22,23);(4,6,7,5)(8,10,11,9)(12,14,15,13)(16,18,19,22)(17,21,20,23)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6806);