Finite Group DatabasebetaFinite Group Databasebeta| order | 5040 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×D3×D5×AGL(1,𝔽7) |
| Fitting, socle | C2×C3×C5×C7 ⋊ C23×C3 |
| derived | C3×C5×C7 ⋊ C24×C3 |
| Hall | C3×C5×ΓL(1,𝔽8) ⋊ C24 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C23×D5 |
| Hall | C5×C7 ⋊ C23×C3×D3 |
| Sylow | C7 ⋊ C22×C3×D3×D5 |
| Sylow | C5 ⋊ C22×D3×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×D5×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C5×C7 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C23×C3 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
1136 conjugacy classes of subgroups.
168 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×AGL(1,𝔽5)×AGL(1,𝔽7), D32×D5×AGL(1,𝔽7), D5×AGL(1,𝔽7)×GL(2,𝔽3) |
| maximal normal subgroup of | C2×D3×AGL(1,𝔽5)×AGL(1,𝔽7) |
| maximal quotient of | D32×D5×AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id.