Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13321 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
552 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)(8,9,14,20)(10,21,24,36)(11,15,28,32)(12,19,13,18)(16,29,34,23)(17,27,31,25)(22,35,26,33)(30,38,37,39);(4,5)(6,7)(9,15)(11,25)(12,13)(16,30)(17,19)(18,31)(20,32)(21,35)(22,38)(23,29)(26,39)(27,28)(33,36)(34,37);(5,7)(8,10)(9,16)(11,26)(12,29)(13,23)(14,24)(15,30)(17,35)(18,36)(19,21)(20,34)(22,28)(25,39)(27,38)(31,33)(32,37)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6309);