Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#13932 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5,6,7)(9,12)(13,15)(16,17,20,18)(19,22,23,21)(25,27)(26,30);(9,12)(13,15)(17,18)(21,22)(24,25,28,27)(26,31,30,29);(8,9,11,12)(10,13,14,15)(16,18,20,17)(19,22,23,21)(24,26,28,30)(25,29,27,31);(8,10,11,14)(9,13,12,15)(16,19)(17,21)(18,22)(20,23)(24,27,28,25)(26,31,30,29)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6920);