Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,14,20)(10,21,25,34)(11,15,28,31)(12,19,13,18)(16,22,35,24)(17,27,30,26)(23,33,38,32)(29,37,36,39);(9,15)(10,22)(11,26)(12,13)(16,32)(17,18)(19,30)(20,31)(21,36)(23,38)(24,25)(27,28)(29,34)(33,35);(5,7)(8,10)(9,16)(11,23)(12,22)(13,24)(14,25)(15,29)(17,33)(18,34)(19,21)(20,35)(26,37)(27,39)(28,38)(30,32)(31,36)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6302);