Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D9⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic9⋊(C22⋊C9) |
| unambiguous semidirect product | (C9⋊(C22⋊C9))⋊C4 |
| derived | C22×C9 ⋊ C4×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D9⋊A4 |
| commutator, hypocenter | C22×C9 | C4×C9 |
| Frattini | C2×C32 | D3×A4 |
| Fitting, cosocle | C23×C3×C9 | C2×C3 |
| socle, prime radical | C23×C32 | C3×D3 |
| LCS | G ▷ C22×C9 | |
| UCS | C1 ◁ C2×C3 | |
120 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3,2,3,2^2,3^2|a^2=c,b^3=d,g^3=h,g^a=g^2h^2,h^a=h^2,e^b=ef,f^b=e,g^b=gh^296 conjugacy classes — view the full character table.
| maximal subgroup of | #72245131, #72245182 |
| maximal normal subgroup of | #72245182 |
| maximal quotient of | #72245131 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 771);