Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#42993 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1718 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)(4,5)(6,10)(7,8)(9,11)(13,16)(14,18)(20,21,25,23)(22,26,28,30)(24,27,32,31)(29,33,34,35);(5,8)(9,10)(12,13,15,16)(14,17,18,19)(22,28)(24,32)(26,30)(27,31);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,16)(18,19)(20,22,25,28)(21,26,23,30)(24,29,32,34)(27,33,31,35);(4,6)(5,9)(7,11)(8,10)(12,14)(13,17)(15,18)(16,19)(20,23,25,21)(22,26,28,30)(24,31,32,27)(29,33,34,35);(20,24,25,32)(21,27,23,31)(22,29,28,34)(26,33,30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35981);