Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34395 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5,6,7)(9,13)(10,16)(15,20)(17,22)(24,25,28,29)(26,32,33,30)(27,35,36,31)(34,37,39,38);(9,13)(14,19)(15,20)(21,23)(25,29)(26,33)(31,35)(34,39);(8,9,12,13)(10,14,16,19)(11,15,18,20)(17,21,22,23)(24,25)(26,30)(27,31)(28,29)(32,33)(34,37)(35,36)(38,39);(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23)(24,26,28,33)(25,30,29,32)(27,34,36,39)(31,37,35,38);(8,11,12,18)(9,15,13,20)(10,17,16,22)(14,21,19,23)(24,27,28,36)(25,31,29,35)(26,34,33,39)(30,37,32,38)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27383);