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×#37261 |
| 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)(14,15)(16,17)(18,19)(21,24)(22,28)(23,31)(33,35);(13,17)(14,18)(20,21,25,24)(22,26,28,30)(23,32,31,27)(29,35,34,33);(4,5)(6,9)(7,8)(10,11)(12,14,16,18)(13,15,17,19)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,34)(32,35);(12,15)(13,18)(14,17)(16,19)(20,23,25,31)(21,27,24,32)(22,29,28,34)(26,33,30,35);(4,6,7,10)(5,9,8,11)(20,24,25,21)(22,30,28,26)(23,32,31,27)(29,35,34,33)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30249);