Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
412 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)(8,9)(10,22)(11,26)(12,20)(13,29)(14,18)(15,21)(16,23)(17,27)(19,31)(24,35)(25,34)(28,38)(30,33)(32,37)(36,39);(4,5)(6,7)(8,10)(9,16)(11,12)(13,30)(14,23)(15,25)(17,29)(18,34)(19,35)(20,22)(21,26)(24,32)(27,31)(28,39)(33,38)(36,37);(5,7)(8,11,12,10,15,23,14,25)(9,17,18,33,21,36,20,35)(13,30,28,27,32,24,31,39)(16,19,22,37,26,38,34,29)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6381);