Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C3×#33398 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C24 |
| commutator | C22 | C23×C3×C8 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C23×C8 |
| prime radical | C2×C3×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9,11,15,12,16,21,27,13,17,22,28,23,29,33,37)(10,14,19,25,20,26,32,36,18,24,30,34,31,35,38,39);(4,5,6,7);(8,10)(9,14)(11,19)(12,20)(13,18)(15,25)(16,26)(17,24)(21,32)(22,30)(23,31)(27,36)(28,34)(29,35)(33,38)(37,39);(10,18)(14,24)(19,30)(20,31)(25,34)(26,35)(32,38)(36,39)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26386);