Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C24×AGL(1,ℤ/9) |
| unambiguous semidirect product | D9⋊(C24×C3) |
| unambiguous semidirect product | 3−1+2⋊C25 |
| unambiguous semidirect product | (C2×3−1+2)⋊C24 |
| Fitting | C24×3−1+2 ⋊ C2 |
| derived | C9 ⋊ C25×C3 |
| Sylow | 3−1+2 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | AGL(1,ℤ/9) |
| commutator, cosocle, hypocenter | C9 | C25×C3 |
| Frattini | C3 | C24×C3×D3 |
| Fitting | C24×3−1+2 | C2 |
| socle | C24×C3 | C3×D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C24 | |
2311 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4047);