Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | AGL(1,𝔽7)×#5935 |
| Fitting | C7×H(5,𝔽2) ⋊ C2×C3×D3 |
| derived | C7×#997 ⋊ C22×C3 |
| Sylow | C7 ⋊ C2×C3×#5935 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽7)×#1020 |
| commutator, cosocle, hypocenter | C7×#997 | C22×C3 |
| Fitting | C7×H(5,𝔽2) | C2×C3×D3 |
| socle | C2×C7 | C2×C3×#1020 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 | |
1384 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
91 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×SO+(4,𝔽3), C2×AGL(1,𝔽7)×#5935, AGL(1,𝔽7)×#73026 |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×#5935, AGL(1,𝔽7)×#73026 |
| maximal quotient of | C2×AGL(1,𝔽7)×#5935 |
| holomorph of | C7×Q8 |
Past the SmallGroups library coverage — no SmallGroups id.