Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#21785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2308 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,16,20,11)(7,13,8,22)(9,15)(10,14)(12,21,30,23)(17,33)(18,27)(19,32,24,28)(25,31)(26,34)(29,35);(4,6)(5,11)(7,17)(8,18)(9,24)(10,20)(12,29)(13,27)(14,16)(15,32)(19,25)(21,35)(22,33)(23,34)(26,30)(28,31);(4,7,10,8)(5,12,14,30)(6,17,20,18)(9,23,25,21)(11,26,16,29)(13,15,22,31)(19,35,24,34)(27,28,33,32);(4,8,10,7)(5,13,14,22)(6,18,20,17)(9,23,25,21)(11,27,16,33)(12,15,30,31)(19,35,24,34)(26,28,29,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14773);