Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13395 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #2034 |
| prime radical | C3×#3050 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C23×C3×C8 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,18)(8,23)(9,12)(10,16)(11,21)(13,20)(15,30)(17,27)(19,26)(22,34)(24,31)(25,32)(28,35)(29,33);(10,25)(14,30)(26,29)(27,31);(4,6,8,24,9,15,20,17)(5,10,13,29,12,25,23,26)(7,19,21,32,22,33,35,16)(11,27,18,30,28,31,34,14)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6383);