Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#20495 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,7)(12,13,16,19,17,20,25,26)(14,18,22,15,21,24,27,23);(5,7)(8,9,11,10)(12,14)(13,18)(15,19)(16,22)(17,21)(20,24)(23,26)(25,27);(8,10,11,9)(14,21)(15,23)(18,24)(22,27);(12,15,16,24,17,23,25,18)(13,14,19,22,20,21,26,27)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13483);