Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#14669 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1342 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)(9,12)(13,15)(16,17)(18,21)(19,20)(22,23)(24,25,28,29)(26,30,31,27);(4,5,6,7)(10,14)(13,15)(17,20)(21,22)(25,29)(26,31);(8,9,11,12)(10,13,14,15)(16,18)(17,21)(19,23)(20,22)(24,26,28,31)(25,27,29,30);(8,10)(9,13)(11,14)(12,15)(16,17)(18,22)(19,20)(21,23)(24,27)(25,26)(28,30)(29,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7657);