Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#34026 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42 | C23 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| socle | C23×C3 | C25 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42 ◁ G | |
2782 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)(10,13)(11,12)(14,15)(16,17)(18,21)(19,20)(22,23);(4,6,7,5)(9,12)(13,15);(17,20)(21,23);(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,13,12,15)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27014);