Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×C4×#401 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| socle, prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5,6,7);(10,14)(12,15)(17,20)(21,22)(24,25,27,29)(26,30,31,28);(8,9,11,13)(10,15,14,12)(16,17,19,20)(18,22,23,21)(24,25)(26,30)(27,29)(28,31);(8,10)(9,12)(11,14)(13,15)(16,18,19,23)(17,21,20,22)(24,26,27,31)(25,28,29,30)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6854);