Finite Group DatabasebetaFinite Group Databasebeta| order | 3240 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽5)×#4183 |
| Fitting, derived, Hall | C5×AΣL(1,𝔽27) ⋊ C2×C4 |
| Sylow | AΣL(1,𝔽27) ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×#4183 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C5×AΣL(1,𝔽27) | C2×C4 |
| Frattini | C32 | C3⋊D3×AGL(1,𝔽5) |
| socle | C3×C5 | C4×#272 |
| prime radical | D5×#4183 | C2 |
| cosocle | D5×AΣL(1,𝔽27) | C22 |
| LCS | G ▷ C5×AΣL(1,𝔽27) | |
| UCS | C1 | |
320 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 45 |
65 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽5)×#4183, AGL(1,𝔽5)×#4183⋊C3 |
| maximal normal subgroup of | C3×AGL(1,𝔽5)×#4183, AGL(1,𝔽5)×#4183⋊C3 |
| maximal quotient of | C3×AGL(1,𝔽5)×#4183 |
Past the SmallGroups library coverage — no SmallGroups id.