Finite Group DatabasebetaFinite Group Databasebeta| order | 5040 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×ΓL(1,𝔽8)⋊GL(2,𝔽3) |
| Fitting | C5×C7×Q8 ⋊ C3×D3 |
| derived | C7×SL(2,𝔽3) ⋊ C2×C3×C5 |
| Hall | C5×C7 ⋊ C3×GL(2,𝔽3) |
| Sylow | C7 ⋊ C3×C5×GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | A4⋊AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C2×C3×C5 |
| Frattini | C2 | C5×A4⋊AGL(1,𝔽7) |
| Fitting | C5×C7×Q8 | C3×D3 |
| socle | C2×C5×C7 | C3×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 | |
140 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
155 conjugacy classes — view the full character table.
| maximal subgroup of | D5×ΓL(1,𝔽8)⋊GL(2,𝔽3), C5×AGL(1,𝔽7)×GL(2,𝔽3), #72218808 |
| maximal normal subgroup of | D5×ΓL(1,𝔽8)⋊GL(2,𝔽3), C5×AGL(1,𝔽7)×GL(2,𝔽3), #72218808 |
Past the SmallGroups library coverage — no SmallGroups id.