Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10279 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,14,11)(10,18,22,19)(12,17,13,16)(15,21,23,20)(24,25,28,31)(26,32,34,39)(27,35,37,30)(29,36,38,33);(5,7)(8,10,14,22)(9,15,11,23)(12,21,13,20)(16,19,17,18);(8,11,14,9)(10,19,22,18)(12,16,13,17)(15,20,23,21)(24,26,28,34)(25,29,31,38)(27,36,37,33)(30,39,35,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3267);