Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#35912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(5,6)(8,9,12,11)(10,14,15,13)(16,17,20,23)(18,21,19,22);(4,5,7,6)(8,9,12,11)(10,14,15,13)(16,18,20,19)(17,21,23,22);(4,6,7,5)(8,10)(9,13)(11,14)(12,15)(16,17)(18,21)(19,22)(20,23);(8,10)(9,14)(11,13)(12,15)(16,19,20,18)(17,22,23,21);(8,11,12,9)(10,14,15,13)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28900);