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×#37244 |
| 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 | |
1468 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)(7,11)(9,10)(13,16)(17,19)(20,21,25,29)(22,28,24,26)(23,31,33,27)(30,34,32,35);(5,6)(9,10)(20,22,25,24)(21,26,29,28)(23,32,33,30)(27,35,31,34);(4,5,8,6)(7,9,11,10)(12,13)(14,17)(15,16)(18,19)(20,21)(22,26)(23,27)(24,28)(25,29)(30,34)(31,33)(32,35);(4,6,8,5)(7,10,11,9)(20,23,25,33)(21,27,29,31)(22,30,24,32)(26,34,28,35);(4,7)(5,9)(6,10)(8,11)(12,14,15,18)(13,17,16,19)(20,24,25,22)(21,28,29,26)(23,32,33,30)(27,35,31,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30232);