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×#37259 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1902 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)(8,9)(10,16)(11,18)(12,20)(13,14)(15,21)(17,19)(22,23)(25,27)(30,32)(31,35)(37,38);(9,14)(11,16)(15,20)(19,22)(24,25,29,27)(26,32,34,30)(28,31,36,35)(33,38,39,37);(4,5)(6,7)(8,10)(9,11)(12,17)(13,18)(14,16)(15,19)(20,22)(21,23)(24,26,29,34)(25,30,27,32)(28,33,36,39)(31,37,35,38);(8,11)(9,10)(12,19)(13,16)(14,18)(15,17)(20,23)(21,22)(24,27,29,25)(26,32,34,30)(28,35,36,31)(33,38,39,37);(8,12,13,21)(9,15,14,20)(10,17,18,23)(11,19,16,22)(24,28,29,36)(25,31,27,35)(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, 30247);