Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#20566 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
974 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)(10,14)(11,13)(12,15)(16,17,21,19,20,23,29,28)(18,24,27,31,26,22,30,25);(4,5,6,7)(8,10)(9,12)(11,15)(13,14)(16,18)(17,22)(19,25)(20,26)(21,27)(23,24)(28,31)(29,30);(10,15)(12,14)(16,19,21,23,20,28,29,17)(18,25,27,24,26,31,30,22);(9,13)(12,14)(18,26)(22,24)(25,31)(27,30)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13554);