Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#20986 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×Q8 ◁ G | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,14,20,27)(11,15)(12,16)(13,17)(18,25,33,37)(19,26,35,39)(21,28)(22,29)(23,30)(24,31)(32,36,34,38);(8,10)(9,14,17,27)(11,18)(12,19)(13,20)(15,25,29,37)(16,26,31,39)(21,32)(22,33)(23,34)(24,35)(28,36,30,38);(5,7)(8,11,13,22)(9,15,17,29)(10,18,20,33)(12,23,24,21)(14,25,27,37)(16,30,31,28)(19,34,35,32)(26,38,39,36);(8,12,13,24)(9,16,17,31)(10,19,20,35)(11,21,22,23)(14,26,27,39)(15,28,29,30)(18,32,33,34)(25,36,37,38)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13974);