Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#16702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9,12,13)(10,14,15,11)(17,18)(21,22)(24,25)(26,30)(27,28)(29,31);(9,13)(10,15)(16,17,20,18)(19,21,23,22)(24,26)(25,29)(27,30)(28,31);(4,5,6,7)(8,10)(9,14)(11,13)(12,15)(16,18,20,17)(19,21,23,22)(24,26)(25,30)(27,29)(28,31);(8,11,12,14)(9,15,13,10)(16,19,20,23)(17,21,18,22)(24,27,28,25)(26,30,31,29)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9690);