Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#10988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18)(11,22,24,15)(12,25)(13,17)(14,20)(16,28)(19,36,38,35)(21,33)(23,32,27,34)(26,30)(29,39,31,37);(8,10,13,21)(9,14,17,30)(11,19,24,38)(12,26,28,20)(15,31,22,29)(16,33,25,18)(23,39,27,37)(32,36,34,35);(5,7)(8,11)(9,15)(10,19)(12,27)(13,24)(14,29)(16,34)(17,22)(18,35)(20,39)(21,38)(23,28)(25,32)(26,37)(30,31)(33,36)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3976);