Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | 3−1+2⋊C32 |
| unambiguous semidirect product | (C9⋊C32)⋊C3 |
| derived | C9 ⋊ C3×C32 |
| Sylow | 3−1+2 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C3×C32 |
| Frattini | C3×C16 | C3×D3 |
| Fitting | C16×3−1+2 | C2 |
| socle | C2×C3 | C3×C3⋊C16 |
| prime radical | C2×C32 | C3⋊C16 |
| cosocle | C9×C16 | C2×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C16 | |
41 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2^4,3^2|a^2=c,c^2=d,d^2=e,e^2=f,g^3=h,g^a=g^2h^2,h^a=h^2,g^b=gh160 conjugacy classes — view the full character table.
| maximal subgroup of | C32×AGL(1,ℤ/9), #10588124⋊C3, C2×3−1+2⋊C32, … |
| maximal normal subgroup of | C32×AGL(1,ℤ/9), #10588124⋊C3, C2×3−1+2⋊C32, … |
| maximal quotient of | 3−1+2⋊C64, C2×3−1+2⋊C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 67);