Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C24×C3 |
| unambiguous semidirect product | C9⋊(C22⋊(C22⋊C9)) |
| Sylow | C24 ⋊ C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C3×C22⋊A4 |
| commutator | C24×C3 | C3×C9 |
| Fitting | C24×C3×C9 | C3 |
| socle, prime radical, cosocle | C24×C32 | C32 |
| hypercenter | C3×C9 | C22⋊A4 |
| hypocenter | C24 | C9⋊C9 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C32 ◁ C3×C9 | |
319 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
3^4,2^4|a^3=d,b^3=c,b^a=bc,e^a=ef,f^a=e,g^a=gh,h^a=g168 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊A4×C9⋊C9, #72229244, #72229255, … |
| maximal normal subgroup of | C22⋊A4×C9⋊C9, #72229244, #72229255, … |
| maximal quotient of | C2×C9⋊(C22⋊(C22⋊C9)), #72276092 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1888);