Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×C3×AΣL(1,𝔽16) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #99 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C2×D4 | C24×C3 |
| prime radical | C22×C3×#120 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3 ◁ C23×C3×D4 ◁ G | |
2774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(13,14,15);(3,4);(5,6)(7,11,12,9)(8,10);(6,9)(10,11)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26329);