Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C3×#20343 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
2130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,5,6,7)(8,9)(10,14)(11,13)(12,16)(15,19)(17,21)(18,20)(22,23);(9,13,16,15)(14,20,21,22);(8,10)(9,14)(11,18)(12,17)(13,20)(15,22)(16,21)(19,23);(10,17)(14,21)(18,23)(20,22)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13331);