Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#20867 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×#120 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C2×#385 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,14,10,13,8,24)(6,15,17,11,19,27,18,26)(7,20,22,12,23,29,21,28)(16,31,32,30,33,25,34,35);(4,6,9,17,10,19,8,18)(5,11,14,27,13,26,24,15)(7,16,22,32,23,33,21,34)(12,25,29,35,28,31,20,30);(4,7)(5,12)(6,16)(8,21)(9,22)(10,23)(11,25)(13,28)(14,29)(15,30)(17,32)(18,34)(19,33)(20,24)(26,31)(27,35);(5,13)(11,26)(12,28)(14,24)(15,27)(20,29)(25,31)(30,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13855);