Finite Group DatabasebetaFinite Group Databasebeta| order | 6480 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×AGL(1,𝔽5)×#1113 |
| Fitting, socle, Hall | C34×C5 ⋊ C22×C4 |
| derived | C33×C5 ⋊ C22×C3×C4 |
| Sylow | C34 ⋊ C22×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C3×C4×#1113 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | AGL(1,𝔽5)×#1113 |
| commutator, hypocenter | C33×C5 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C34×C5 | C22×C4 |
| prime radical | C3×D5×#1113 | C2 |
| cosocle | C33×D5 | C23×C3 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 ◁ C3 | |
1548 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1548 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
225 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×(((C3⋊D3)⋊C9)⋊C2)⋊C3, AGL(1,𝔽5)×#10620215, AGL(1,𝔽5)×#11757730 |
| maximal normal subgroup of | AGL(1,𝔽5)×#10620215 |
| maximal quotient of | AGL(1,𝔽5)×(#4207⋊C2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id.