Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| direct product | D3×AGL(3,𝔽2) |
| Fitting, socle | C23×C3 ⋊ C2×GL(3,𝔽2) |
| derived | C3×AGL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×AGL(3,𝔽2) | C2 |
| Fitting, socle, cosocle | C23×C3 | C2×GL(3,𝔽2) |
| LCS | G ▷ C3×AGL(3,𝔽2) | |
| UCS | C1 | |
786 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 786 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×AGL(3,𝔽2), S4×AGL(3,𝔽2) |
| maximal normal subgroup of | C2×D3×AGL(3,𝔽2) |
| maximal quotient of | C2×D3×AGL(3,𝔽2), S4×AGL(3,𝔽2), Dic3×AGL(3,𝔽2) |
| automorphism group of | C23×D3 |
| holomorph of | C23×C3 |
Past the SmallGroups library coverage — no SmallGroups id.