Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#52030 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1556 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)(12,13,16,14)(15,18,19,17)(20,21,25,23)(22,28,30,26)(24,31,32,27)(29,33,35,34);(4,5)(6,8)(7,10)(9,11)(12,14,16,13)(15,18,19,17)(21,23)(26,28)(27,31)(33,34);(5,6)(9,10)(12,15)(13,17)(14,18)(16,19)(20,22,25,30)(21,26,23,28)(24,29,32,35)(27,33,31,34);(4,6,8,5)(7,10,11,9)(20,23,25,21)(22,28,30,26)(24,31,32,27)(29,34,35,33);(4,7,8,11)(5,9,6,10)(20,24)(21,27)(22,29)(23,31)(25,32)(26,33)(28,34)(30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45018);