Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#36846 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(6,10)(9,11)(13,16)(17,19)(20,21,25,28)(22,29,23,26)(24,32,33,27)(30,34,31,35);(5,8)(9,11)(20,22,25,23)(21,26,28,29)(24,30,33,31)(27,34,32,35);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,16)(18,19)(20,23,25,22)(21,26,28,29)(24,31,33,30)(27,34,32,35);(4,5)(6,9)(7,8)(10,11)(20,21)(22,29)(23,26)(24,27)(25,28)(30,35)(31,34)(32,33);(4,6)(5,9)(7,10)(8,11)(12,14,15,18)(13,17,16,19)(20,24)(21,27)(22,30)(23,31)(25,33)(26,34)(28,32)(29,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29834);