Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C22×C33 |
| direct product | C3×(C22⋊A4)⋊C9 |
| derived, Sylow | C24 ⋊ C32×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | A42 |
| commutator, hypocenter | C24 | C32×C9 |
| Frattini | C3 | C3×A42 |
| Fitting, socle | C24×C32 | C32 |
| prime radical | C32×C22⋊A4 | C3 |
| cosocle | C24×C3 | C33 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C32 | |
308 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#13049240, #72229281, C3×#11724027, … |
| maximal normal subgroup of | C3×#13049240, #72229281, C3×#11724027, … |
| maximal quotient of | #72229223, A4×(C22⋊A4)⋊C9, C2×C3×(C22⋊A4)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2956);