Finite Group DatabasebetaFinite Group Databasebeta| order | 3024 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×D9×AGL(1,𝔽7) |
| Fitting | C22×C7×C9 ⋊ C22×C3 |
| derived | C7×C9 ⋊ C24×C3 |
| Hall | C9×ΓL(1,𝔽8) ⋊ C24 |
| Sylow | C7 ⋊ C23×C3×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D9×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7×C9 | C24×C3 |
| Frattini | C3 | C22×D3×AGL(1,𝔽7) |
| Fitting | C22×C7×C9 | C22×C3 |
| socle | C22×C3×C7 | C2×C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C22 | |
868 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C22×AGL(1,𝔽7)×AGL(1,ℤ/9) |
| maximal normal subgroup of | C22×AGL(1,𝔽7)×AGL(1,ℤ/9) |
Past the SmallGroups library coverage — no SmallGroups id.