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) = C23×C4 |
| direct product | C3×#20559 |
| 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 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,9)(10,14)(11,13)(12,15)(16,17,20,18,19,21,23,22);(4,5,6,7)(8,10)(9,12)(11,15)(13,14)(17,21)(18,22);(10,15)(12,14)(16,18,20,21,19,22,23,17);(9,13)(12,14)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13547);