Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#42961 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2619 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(12,13,16,18)(14,15,19,17)(20,21,25,23)(22,26,28,30)(24,27,32,31)(29,33,34,35);(5,6)(9,10)(12,14,16,19)(13,17,18,15)(22,28)(24,32)(26,30)(27,31);(4,6,8,5)(7,10,11,9)(12,14)(13,15)(16,19)(17,18)(20,22)(21,26)(23,30)(24,29)(25,28)(27,33)(31,35)(32,34);(4,7)(5,9)(6,10)(8,11)(12,15)(13,14)(16,17)(18,19)(20,23,25,21)(22,26,28,30)(24,31,32,27)(29,33,34,35);(20,24,25,32)(21,27,23,31)(22,29,28,34)(26,33,30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35949);