Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13363 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(8,9,11,13)(10,14,15,12)(16,17)(18,21)(19,28)(20,24)(22,25)(23,26)(27,30)(29,31);(4,5)(6,7)(8,10,11,15)(9,12,13,14);(5,7)(17,21)(18,25)(19,26)(22,23)(24,30)(27,31)(28,29)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6351);