Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#47851 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5,6,7)(17,21)(19,22);(8,9)(10,14)(11,15)(12,13)(16,17,20,21)(18,19,23,22);(4,5)(6,7)(9,13)(10,15)(16,18,20,23)(17,22,21,19);(8,10)(9,11)(12,15)(13,14)(16,18)(17,19)(20,23)(21,22);(8,11)(9,10)(12,14)(13,15)(16,19)(17,18)(20,22)(21,23)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 40839);