Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D9⋊(Q8⋊C9) |
| derived | Q8×C9 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D9⋊A4 |
| commutator, hypocenter | Q8×C9 | C2×C9 |
| Frattini, socle | C2×C32 | D3×A4 |
| Fitting, cosocle | C3×Q8×C9 | C2×C3 |
| prime radical | C2×C3×D9 | A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2×C3 | |
101 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^2,3,2,3|b^3=c,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^b=fh^2,e^d=eg78 conjugacy classes — view the full character table.
| maximal subgroup of | #72252993, #72253007 |
| maximal normal subgroup of | #72253007 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 758);