Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60127 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ G | |
2168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27);(5,8)(9,11);(4,6,7,10)(5,9,8,11);(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(14,20)(17,23)(21,26)(24,27)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53115);