Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#33087 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,11)(7,9)(8,10)(12,13)(14,16)(15,17)(18,19)(20,21)(22,31)(23,28)(24,34)(25,30)(26,27)(29,32)(33,35);(4,6,8,7)(5,9,10,11)(12,14,15,18)(13,16,17,19)(20,22,25,26)(21,27,30,31)(23,33,24,32)(28,29,34,35);(4,7,8,6)(5,9,10,11)(13,17)(16,19)(20,23,25,24)(21,28,30,34)(22,32,26,33)(27,35,31,29);(20,24,25,23)(21,29,30,35)(22,32,26,33)(27,34,31,28)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26075);