Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #88614⋊C3 |
| unambiguous semidirect product | #88615⋊C3 |
| unambiguous semidirect product | #88724⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×AGL(1,ℤ/9) |
| commutator, cosocle, hypocenter | Q8×C9 | C2×C32 |
| Frattini, socle | C2×C3 | C3×D3×A4 |
| Fitting | Q8×3−1+2 | C2×C3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2 | |
151 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^2,3,2,3|a^2=g,d^2=g,e^2=g,f^3=h,f^a=f^2h^2,h^a=h^2,d^b=de,e^b=d,f^c=fh^2,e^d=eg70 conjugacy classes — view the full character table.
| maximal subgroup of | #72245909, #72253022 |
| maximal quotient of | #72253019 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2154);