Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13332 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
566 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)(10,20)(11,16)(12,18)(13,29)(14,19)(15,22)(17,26)(21,30)(23,35)(24,37)(25,36)(27,33)(28,34)(31,39)(32,38);(5,7)(8,10,14,24)(9,15,19,32)(11,25,28,39)(12,21,13,23)(16,20,34,37)(17,35,33,30)(18,36,29,31)(22,27,38,26);(9,16)(10,21)(11,26)(12,13)(15,30)(17,18)(19,34)(20,36)(22,38)(23,24)(27,28)(29,33)(31,37)(32,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6320);