Finite Group DatabasebetaFinite Group Databasebeta| order | 7320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×AGL(1,𝔽61) |
| Fitting, socle | C2×C61 ⋊ C3×C4×C5 |
| derived, Sylow | C61 ⋊ C2×C3×C4×C5 |
| Hall | C2×#6923⋊C5 ⋊ C3 |
| Hall | C2×#6923⋊C3 ⋊ C5 |
| Hall | (C61⋊C3)⋊C5 ⋊ C2×C4 |
| Hall | C2×#6923 ⋊ C3×C5 |
| Hall | C61⋊C5 ⋊ C2×C3×C4 |
| Hall | C61⋊C3 ⋊ C2×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽61) |
| commutator, hypocenter | C61 | C2×C3×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C61 | C3×C4×C5 |
| prime radical | C2×(D61⋊C3)⋊C5 | C2 |
| cosocle | D61 | C22×C3×C5 |
| LCS | G ▷ C61 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 122 |
122 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AGL(1,𝔽61), C22×AGL(1,𝔽61) |
| maximal normal subgroup of | C22×AGL(1,𝔽61) |
| maximal quotient of | D3×AGL(1,𝔽61), C22×AGL(1,𝔽61) |
| automorphism group of | Dic61, C2×D61 |
| holomorph of | C2×C61 |
Past the SmallGroups library coverage — no SmallGroups id.