Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | ((C3⋊S4)⋊C9)⋊C2 |
| unambiguous semidirect product | #10619754⋊C2 |
| unambiguous semidirect product | #10619770⋊C2 |
| Fitting | C22×C3×C9 ⋊ C2×D3 |
| derived | (C3×A4)⋊C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | (C3×A4)⋊C9 | C22 |
| Frattini | C32 | D3×S4 |
| Fitting | C22×C3×C9 | C2×D3 |
| socle | C22×C32 | D32 |
| LCS | G ▷ (C3×A4)⋊C9 | |
| UCS | C1 | |
220 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3^4,2^2|d^3=e,d^a=d^2e^2,e^a=e^2,f^a=f^2,c^b=c^2,f^b=f^2,g^b=gh,d^c=df,g^c=h,h^c=gh39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1785);