Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#16696 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1122 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)(5,7)(8,9,12,10)(11,14,15,13)(17,18)(21,22)(26,30)(28,31);(9,10)(13,14)(16,17,20,18)(19,21,23,22)(24,25,27,29)(26,28,30,31);(4,5)(6,7)(8,10,12,9)(11,13,15,14)(16,18,20,17)(19,21,23,22);(8,11,12,15)(9,13,10,14)(16,19,20,23)(17,21,18,22)(24,26)(25,28)(27,30)(29,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9684);