Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | D7×Ω+(4,𝔽3) |
| Fitting, derived | C7×H(5,𝔽2) ⋊ C2×C32 |
| Hall | D7×H(5,𝔽2) ⋊ C32 |
| Sylow | C7 ⋊ C2×Ω+(4,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A42×D7 |
| commutator, Fitting, cosocle, hypocenter | C7×H(5,𝔽2) | C2×C32 |
| socle | C2×C7 | C2×A42 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
316 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
125 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×Ω+(4,𝔽3), D7×SO+(4,𝔽3) |
| maximal normal subgroup of | AGL(1,𝔽7)×Ω+(4,𝔽3), D7×SO+(4,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.