Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C7×SO+(4,𝔽3) |
| Fitting | C7×H(5,𝔽2) ⋊ C3⋊D3 |
| derived | Ω+(4,𝔽3) ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C7 | A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C2×C7 |
| Frattini | C2 | C7×A4⋊S4 |
| Fitting | C7×H(5,𝔽2) | C3⋊D3 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2×C7 | |
254 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×SO+(4,𝔽3), D7×SO+(4,𝔽3) |
| maximal normal subgroup of | ΓL(1,𝔽8)×SO+(4,𝔽3), D7×SO+(4,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.