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×#37262 |
| 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 | |
1476 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)(5,8)(9,11)(12,13,16,14)(15,18,19,17)(20,21)(22,27)(23,26)(24,28)(25,30)(29,34)(31,33)(32,35);(13,14)(17,18)(20,22,26,25)(21,27,23,30)(24,32,33,29)(28,35,31,34);(4,5)(6,9)(7,8)(10,11)(12,14,16,13)(15,18,19,17)(20,23,26,21)(22,27,25,30)(24,31,33,28)(29,34,32,35);(12,15)(13,17)(14,18)(16,19)(20,24,26,33)(21,28,23,31)(22,29,25,32)(27,34,30,35);(4,6,7,10)(5,9,8,11)(20,25,26,22)(21,27,23,30)(24,32,33,29)(28,34,31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30250);