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) = C22×C42 |
| direct product | C3×#20481 |
| 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 | C24×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
962 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)(8,9)(10,13)(11,12)(14,15)(16,17,20,24,21,19,23,28)(18,22,27,30,26,25,29,31);(4,5,6,7)(9,12)(13,15)(16,18)(17,22)(19,25)(20,27)(21,26)(23,29)(24,30)(28,31);(16,19,20,28,21,17,23,24)(18,25,27,31,26,22,29,30);(8,10,11,14)(9,13,12,15)(17,19)(18,26)(24,28)(27,29)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13469);