Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#37121 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2002 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)(5,7)(11,18)(15,20)(17,22)(21,23)(24,25)(26,32)(27,31)(28,29)(30,34)(33,38)(35,36)(37,39);(9,13)(10,16)(15,20)(17,22)(25,29)(30,32)(31,36)(37,38);(4,5)(6,7)(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23)(24,25)(26,30)(27,31)(28,29)(32,34)(33,37)(35,36)(38,39);(8,10,12,16)(9,14,13,19)(11,17,18,22)(15,21,20,23)(24,26,28,34)(25,30,29,32)(27,33,35,39)(31,37,36,38);(8,11)(9,15)(10,17)(12,18)(13,20)(14,21)(16,22)(19,23)(24,27,28,35)(25,31,29,36)(26,33,34,39)(30,37,32,38)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30109);