Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D3×AGL(1,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊AGL(1,ℤ/9))⋊C4 |
| unambiguous semidirect product | ((C3⋊D9)⋊C4)⋊C3 |
| derived | C3×C9 ⋊ C2×C3×C4 |
| Sylow | C3×3−1+2 ⋊ C2×C4 |
135 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3,2,3^3|a^2=d,b^2=d,e^3=g,f^a=f^2,e^b=e^2g^2,g^b=g^2,e^c=eg^260 conjugacy classes — view the full character table.
| maximal subgroup of | (C3⋊AGL(1,ℤ/9))⋊Q8, #88635⋊C3, #88642⋊C3, … |
| maximal normal subgroup of | (C3⋊AGL(1,ℤ/9))⋊Q8, #88635⋊C3, #88642⋊C3, … |
| maximal quotient of | (C3⋊AGL(1,ℤ/9))⋊C8, #88414⋊C3, Dic3×3−1+2⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 381);