Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽7)×GL(3,𝔽2) |
| Fitting | C7 ⋊ C2×C3×GL(3,𝔽2) |
| socle, derived | C7×GL(3,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C7×GL(3,𝔽2) | C2×C3 |
| Fitting, cosocle | C7 | C2×C3×GL(3,𝔽2) |
| LCS | G ▷ C7×GL(3,𝔽2) | |
| UCS | C1 | |
182 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
42 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×AGL(3,𝔽2) |
| maximal quotient of | AGL(1,𝔽7)×AGL(3,𝔽2) |
| automorphism group of | C23×ΓL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id.