Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (C22⋊C9)⋊(C22⋊C9) |
| Sylow | C24 ⋊ C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | A42 |
| commutator | C24×C3 | C3×C9 |
| Fitting, socle, prime radical, cosocle | C24×C32 | C32 |
| hypocenter | C24 | C9⋊C9 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C32 | |
158 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 162 |
Polycyclic presentation:
3^4,2^4|a^3=d,b^3=c,b^a=bd,g^a=gh,h^a=g,e^b=ef,f^b=e96 conjugacy classes — view the full character table.
| maximal subgroup of | #72220143, #72220166, #72229281, … |
| maximal normal subgroup of | #72220143, #72220166, #72229281, … |
| maximal quotient of | #72229255, C2×(C22⋊C9)⋊(C22⋊C9), #72244993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1798);