Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2286 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)(4,5,7,6)(9,12)(10,14)(16,17,21,19)(18,22,26,24)(20,27,28,23)(25,30,31,29);(9,12)(13,15)(17,19)(22,24)(23,27)(29,30);(4,6,7,5)(8,9)(10,13)(11,12)(14,15)(16,18,21,26)(17,22,19,24)(20,25,28,31)(23,29,27,30);(8,10)(9,13)(11,14)(12,15)(16,19,21,17)(18,24,26,22)(20,27,28,23)(25,30,31,29);(16,20,21,28)(17,23,19,27)(18,25,26,31)(22,29,24,30)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27401);