Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18282 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#1600 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
942 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)(5,10)(6,14)(7,18)(11,23)(12,21)(13,19)(17,28)(20,29)(22,24)(25,32)(26,34)(33,35);(4,5,8,13)(6,11,14,24)(7,19,18,10)(9,12,15,25)(16,23,27,22)(17,26,28,35)(20,32,29,21)(30,33,31,34);(6,14)(7,20)(10,21)(11,24)(16,30)(17,28)(18,29)(19,32)(22,34)(23,33)(26,35)(27,31);(4,6,15,28)(5,11,25,35)(7,16,29,30)(8,14,9,17)(10,22,32,34)(12,26,13,24)(18,27,20,31)(19,23,21,33)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11270);