Finite Group DatabasebetaFinite Group Databasebeta| order | 3024 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D3×#4247⋊C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C2×D4 |
| Sylow | C7 ⋊ C3×D3×D12 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×AGL(1,𝔽7) |
| commutator, Fitting, socle, cosocle | C2×C32×C7 | C23×C3 |
| hypocenter | C32×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
540 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | D32×(C22⋊D7)⋊C3, D3×#66694⋊C3, D3×D12×AGL(1,𝔽7), … |
| maximal normal subgroup of | D32×(C22⋊D7)⋊C3, D3×#66694⋊C3, D3×D12×AGL(1,𝔽7), … |
Past the SmallGroups library coverage — no SmallGroups id.