Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#34346 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
2442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,24)(19,23)(20,22)(21,25)(26,30)(27,29)(28,31);(10,15)(12,14)(16,18)(17,21)(19,26)(20,25)(22,24)(23,28)(27,31)(29,30);(8,10,11,15)(9,12,13,14)(18,25)(21,24)(26,31)(28,30);(9,13)(12,14)(17,22)(18,25)(23,29)(26,31);(16,19,20,27)(17,23,22,29)(18,26,25,31)(21,28,24,30)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27334);