Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13389 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2033 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
402 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)(5,11)(6,15)(7,9)(8,21)(12,27)(13,30)(14,19)(16,34)(17,22)(18,33)(20,23)(24,32)(25,28)(29,31);(4,5)(6,16)(7,19)(8,13)(9,11)(10,14)(12,18)(15,31)(17,25)(20,35)(21,26)(22,34)(23,30)(24,29)(27,32)(28,33);(4,6,7,17,10,18,9,24)(5,12,11,25,14,16,19,29)(8,22,20,15,23,32,21,33)(13,27,26,31,30,34,35,28)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6377);