Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#42978 |
| 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 | |
1904 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)(8,9,13,12)(10,16,18,14)(11,19,21,15)(17,22,20,23)(24,25,29,32)(26,30,34,28)(27,31,37,39)(33,38,35,36);(9,12)(14,16)(15,19)(22,23)(24,26,29,34)(25,30,32,28)(27,33,37,35)(31,38,39,36);(4,5)(6,7)(8,10,13,18)(9,14,12,16)(11,20,21,17)(15,23,19,22)(24,26)(25,30)(27,35)(28,32)(29,34)(31,36)(33,37)(38,39);(8,11,13,21)(9,15,12,19)(10,17,18,20)(14,22,16,23)(24,27,29,37)(25,31,32,39)(26,33,34,35)(28,36,30,38);(8,12,13,9)(10,16,18,14)(11,19,21,15)(17,23,20,22)(24,28,29,30)(25,26,32,34)(27,36,37,38)(31,33,39,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35966);