Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#34195 |
| 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 | |
2370 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)(4,5,6,7)(9,12)(13,15)(16,17)(18,24)(19,27)(20,21)(22,26)(23,28)(25,29)(30,31);(17,21)(22,24)(23,27)(29,30);(8,9)(10,13)(11,12)(14,15)(16,18,20,26)(17,22,21,24)(19,25,28,31)(23,29,27,30);(16,17)(18,22)(19,23)(20,21)(24,26)(25,29)(27,28)(30,31);(8,10,11,14)(9,13,12,15)(16,19)(17,23)(18,25)(20,28)(21,27)(22,29)(24,30)(26,31)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27183);