Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#52164 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
2716 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)(8,9,11,12)(10,13,14,15)(16,17,19,21)(18,22,23,20);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,18)(17,20)(19,23)(21,22);(5,7)(17,21)(20,22);(9,12)(13,15)(18,23)(20,22);(8,10,11,14)(9,13,12,15)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45152);