Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊(((C22⋊C9)⋊C2)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊(#1094⋊C4) |
| derived | C3×C22⋊C9 ⋊ C3×C4 |
272 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3,2,3^3,2^2|a^2=c,d^3=f,d^a=d^2f^2,e^a=e^2,f^a=f^2,g^a=gh,d^b=df,g^d=h,h^d=gh78 conjugacy classes — view the full character table.
| maximal subgroup of | #72220152, Dic3×((C22⋊C9)⋊C2)⋊C3, #72220188, … |
| maximal normal subgroup of | #72220152, Dic3×((C22⋊C9)⋊C2)⋊C3, #72220188, … |
| maximal quotient of | C2×C3⋊(#1094⋊C4), #72252951 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2129);