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×#47845 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2540 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)(5,7)(8,9,13,15)(10,12,11,14);(8,10,13,11)(9,14,15,12)(16,17)(18,22)(19,23)(20,21);(4,5)(6,7)(9,15)(10,11)(17,21)(18,23);(8,11,13,10)(9,14,15,12)(16,18)(17,19)(20,23)(21,22);(8,12,13,14)(9,10,15,11)(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, 40833);