Finite Group DatabasebetaFinite Group Databasebeta| order | 6720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D5×D7×GL(2,𝔽3) |
| Fitting | C5×C7×Q8 ⋊ C22×D3 |
| derived | C5×C7×SL(2,𝔽3) ⋊ C23 |
| Hall | C5×C7 ⋊ C22×GL(2,𝔽3) |
| Sylow | C7 ⋊ C2×D5×GL(2,𝔽3) |
| Sylow | C5 ⋊ C2×D7×GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×D7×S4 |
| commutator, cosocle, hypocenter | C5×C7×SL(2,𝔽3) | C23 |
| Fitting | C5×C7×Q8 | C22×D3 |
| socle | C2×C5×C7 | C22×S4 |
| LCS | G ▷ C5×C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
852 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
160 conjugacy classes — view the full character table.
| maximal subgroup of | D5×AGL(1,𝔽7)×GL(2,𝔽3) |
| maximal normal subgroup of | D5×AGL(1,𝔽7)×GL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.