Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G | |
820 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)(5,6)(8,9)(10,15)(11,13)(12,14)(16,17)(18,24)(19,22)(20,23)(21,26)(25,30)(27,29)(28,31);(4,5,7,6)(8,10,12,11)(9,13,14,15)(16,18)(17,21)(19,26)(20,25)(22,24)(23,29)(27,28)(30,31);(4,5,7,6)(8,11,12,10)(9,13,14,15)(18,25)(20,28)(21,29)(23,31)(24,30)(26,27);(4,6,7,5)(9,14)(13,15)(17,22)(21,24)(23,31)(29,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19778);