Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×AGL(1,𝔽7)×GO+(4,𝔽2) |
| Fitting, socle | C2×C32×C7 ⋊ C2×C3×D4 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C22×D4 |
| Sylow | C7 ⋊ C22×C3×GO+(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽7)×GO+(4,𝔽2) |
| commutator, cosocle | C7×C3⋊D3 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C2×C3×D4 |
| hypocenter | C32×C7 | C22×C3×D4 |
| LCS | G ▷ C7×C3⋊D3 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
1608 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1608 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
126 conjugacy classes — view the full character table.
| maximal subgroup of | C4×AGL(1,𝔽7)×GO+(4,𝔽2), C2×AGL(1,𝔽7)×AΓL(1,𝔽9) |
| maximal normal subgroup of | C4×AGL(1,𝔽7)×GO+(4,𝔽2), C2×AGL(1,𝔽7)×AΓL(1,𝔽9) |
| maximal quotient of | C4×AGL(1,𝔽7)×GO+(4,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id.