Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#21777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2308 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C24×C3 ◁ G | |
1318 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)(6,14,17,10)(7,11)(8,20,21,12)(9,13)(15,22,29,24)(16,23)(18,31,32,25)(19,26)(27,30)(28,34)(33,35);(4,6)(5,10)(7,15)(8,16)(9,17)(11,24)(12,23)(13,14)(18,28)(19,29)(20,27)(21,30)(22,26)(25,34)(31,33)(32,35);(4,7,9,19)(5,11,13,26)(6,15,17,29)(8,18,21,32)(10,22,14,24)(12,25,20,31)(16,28,30,35)(23,33,27,34);(4,8)(5,12)(6,16)(7,18)(9,21)(10,23)(11,25)(13,20)(14,27)(15,28)(17,30)(19,32)(22,33)(24,34)(26,31)(29,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14765);