Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | D3×ΓL(1,𝔽16)⋊D3 |
| Fitting, socle, derived, Hall | C33×C5 ⋊ C22×C4 |
| Sylow | C33 ⋊ C22×AGL(1,𝔽5) |
| Sylow | C5 ⋊ D3×#624 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C33×C5 | C22×C4 |
| prime radical | D3×D5×C3⋊D3 | C2 |
| cosocle | C33×D5 | C23 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 | |
580 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
72 conjugacy classes — view the full character table.
| maximal subgroup of | D33×AGL(1,𝔽5), #72222584, D3×#14268436, … |
| maximal normal subgroup of | D33×AGL(1,𝔽5), C2×D3×ΓL(1,𝔽16)⋊D3 |
| maximal quotient of | #72222584, D3×#14268436, C2×D3×ΓL(1,𝔽16)⋊D3, … |
Past the SmallGroups library coverage — no SmallGroups id.