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×#41429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(13,15)(14,18)(20,21,25,29)(22,26,30,35)(23,27,33,32)(24,28,34,31);(5,8)(9,10)(12,13,16,15)(14,19,18,17)(22,30)(24,34)(26,35)(28,31);(4,5)(6,9)(7,8)(10,11)(12,14,16,18)(13,17,15,19)(20,22)(21,26)(23,28)(24,32)(25,30)(27,34)(29,35)(31,33);(4,6,7,11)(5,9,8,10)(12,15,16,13)(14,19,18,17)(20,23)(21,27)(22,31)(24,26)(25,33)(28,30)(29,32)(34,35);(20,24)(21,28)(22,32)(23,26)(25,34)(27,30)(29,31)(33,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34417);