Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25686 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(8,9,11,12)(10,14,15,13)(16,17,22,24)(18,23,27,31)(19,21,28,20)(25,26,30,29);(4,5)(6,7)(9,12)(13,14)(16,18)(17,23)(19,25)(20,29)(21,26)(22,27)(24,31)(28,30);(5,7)(16,19,21,24,22,28,20,17)(18,25,26,31,27,30,29,23);(8,10,11,15)(9,13,12,14)(16,20,22,21)(17,19,24,28)(18,26,27,29)(23,30,31,25)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18674);