Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | AGL(1,𝔽7)×#1020⋊C2 |
| Fitting | C7×#120 ⋊ C2×C3×D3 |
| derived | C7×C22⋊A4 ⋊ C23×C3 |
| Sylow | C7 ⋊ C2×C3×#1020⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C23×C3 |
| Frattini | C22 | C2×S4×AGL(1,𝔽7) |
| Fitting | C7×#120 | C2×C3×D3 |
| socle | C22×C7 | C22×C3×S4 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 | |
1890 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1890 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.
98 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×AGL(3,𝔽2), #72224594 |
| maximal quotient of | #72224594, AGL(1,𝔽7)×#73026 |
Past the SmallGroups library coverage — no SmallGroups id.