Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×ΓL(1,𝔽8)×D12 |
| unambiguous semidirect product | C4⋊(C3×D3×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C7×D12)⋊C32 |
| unambiguous semidirect product | (C3×C7×D12)⋊C3 |
| Fitting | C32×C4×C7 ⋊ C2×C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ D4 |
| Sylow | C7 ⋊ C32×D12 |
240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#3802, C3×D12×AGL(1,𝔽7), #72246943, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#3802, C3×D12×AGL(1,𝔽7), #72246943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 640);