Finite Group DatabasebetaFinite Group Databasebeta| order | 8640 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | AGL(1,𝔽5)×(C22⋊A4)⋊C9 |
| derived | C24×C5 ⋊ C3×C4×C9 |
| Hall | C24×AGL(1,𝔽5) ⋊ C3×C9 |
| Sylow | C5 ⋊ C4×(C22⋊A4)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A42×AGL(1,𝔽5) |
| commutator, hypocenter | C24×C5 | C3×C4×C9 |
| Fitting, socle | C24×C3×C5 | C32×C4 |
| prime radical | C3×D5×C22⋊A4 | C2×C3 |
| cosocle | C24×C3×D5 | C2×C32 |
| LCS | G ▷ C24×C5 | |
| UCS | C1 ◁ C3 | |
714 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#13049240 |
| maximal normal subgroup of | AGL(1,𝔽5)×#13049240 |
Past the SmallGroups library coverage — no SmallGroups id.