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×#34378 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2010 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)(10,14)(16,17)(18,25)(19,22)(20,28)(21,24)(23,29)(26,30)(27,31);(9,12)(13,15)(16,18,21,19)(17,22,24,25)(20,26,29,27)(23,30,28,31);(8,9)(10,13)(11,12)(14,15)(16,19,21,18)(17,22,24,25)(20,27,29,26)(23,30,28,31);(8,10,11,14)(9,13,12,15)(16,17)(18,25)(19,22)(20,23)(21,24)(26,31)(27,30)(28,29);(16,20)(17,23)(18,26)(19,27)(21,29)(22,30)(24,28)(25,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27366);