Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19081 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
880 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)(5,10)(6,14)(7,8)(12,13)(15,16)(17,18)(20,21,26,23)(22,30,33,35)(24,29,25,28)(27,34,31,32);(4,5)(6,11)(7,12)(8,10)(9,13)(14,18)(15,17)(16,19)(20,22)(21,27)(23,31)(24,32)(25,34)(26,33)(28,30)(29,35);(4,6,9,16)(5,11,13,19)(7,14,8,15)(10,17,12,18)(20,23,26,21)(22,31,33,27)(24,28,25,29)(30,34,35,32);(4,7,9,8)(5,12,13,10)(6,15,16,14)(11,17,19,18)(20,24,26,25)(21,28,23,29)(22,32,33,34)(27,30,31,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12069);