Finite Group DatabasebetaFinite Group Databasebeta| order | 4320 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22×C3 |
| direct product | A4×A6 |
| Fitting | C22 ⋊ C3×A6 |
| socle, derived | C22×A6 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C22×A6 | C3 |
| Fitting, cosocle | C22 | C3×A6 |
| LCS | G ▷ C22×A6 | |
| UCS | C1 | |
156 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
28 conjugacy classes — view the full character table.
| derived subgroup of | S4×A6 |
| maximal subgroup of | A5×A6, S4×A6 |
| maximal normal subgroup of | S4×A6 |
| maximal quotient of | A4×SL(2,𝔽9), SL(2,𝔽3)×A6, #72228176, … |
Past the SmallGroups library coverage — no SmallGroups id.