Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#47805 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1534 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)(14,18)(16,19)(20,21)(22,27)(23,28)(24,32)(25,26)(29,33)(30,31)(34,35);(4,5)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(21,26)(23,27)(29,32)(31,34);(5,8)(9,10)(12,14)(13,16)(15,18)(17,19)(20,22,25,28)(21,27,26,23)(24,30,33,35)(29,34,32,31);(4,5)(6,9)(7,8)(10,11)(20,23,25,27)(21,28,26,22)(24,31,33,34)(29,35,32,30);(4,6,7,11)(5,9,8,10)(20,24)(21,29)(22,30)(23,31)(25,33)(26,32)(27,34)(28,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 40793);