Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C3×#33401 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C22×D4 |
| commutator | C4 | C23×C3×C8 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C22×C4⋊C8 |
| prime radical | C2×C3×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×#131 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,15,10,14,20,30,11,16,19,12,8,25,6,17)(7,13,23,29,24,28,32,35,21,27,33,26,22,34,18,31);(4,6,8,19,11,20,10,9)(5,12,14,17,16,15,25,30)(7,18,22,33,21,32,24,23)(13,26,28,31,27,29,34,35);(4,7)(5,13)(6,18)(8,22)(9,23)(10,24)(11,21)(12,26)(14,28)(15,29)(16,27)(17,31)(19,33)(20,32)(25,34)(30,35);(7,21)(13,27)(18,32)(22,24)(23,33)(26,29)(28,34)(31,35)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26389);