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×#34497 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#553 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5,7,6)(8,9,11,13)(10,14,15,12)(16,17)(18,22)(19,21)(20,23);(8,10,11,15)(9,12,13,14)(16,18,21,20)(17,22,19,23);(8,10)(9,12)(11,15)(13,14)(16,19,21,17)(18,22,20,23);(4,6,7,5)(8,10)(9,12)(11,15)(13,14)(16,20,21,18)(17,22,19,23);(8,9)(10,12)(11,13)(14,15)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27485);