Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C3×C4 |
| direct product | C3⋊Dic3×ΓL(1,𝔽16) |
| derived, Hall | C33×C5 ⋊ C42 |
| Sylow | C33 ⋊ C4×AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic3×C3⋊Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3×ΓL(1,𝔽16) |
| commutator, hypocenter | C33×C5 | C42 |
| Fitting, socle | C2×C33×C5 | C2×C4 |
| prime radical, cosocle | C2×C33×D5 | C22 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 ◁ C2 | |
488 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | #72254802, D3×Dic3×ΓL(1,𝔽16), AGL(1,𝔽5)×#6289 |
| maximal normal subgroup of | #72254802, D3×Dic3×ΓL(1,𝔽16), AGL(1,𝔽5)×#6289 |
Past the SmallGroups library coverage — no SmallGroups id.