Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D3×ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3×Dic3×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×C3)⋊(C3×C4×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C7×Dic3)⋊C32 |
| unambiguous semidirect product | (C2×C3×C7×Dic3)⋊C3 |
| derived | C3×C7 ⋊ C2×C32×C4 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C2×C4 |
| Sylow | C7 ⋊ C2×C32×Dic3 |
272 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×#624, C2×C3×Dic3×AGL(1,𝔽7), C2×#92687⋊C3, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×#624, C2×C3×Dic3×AGL(1,𝔽7), C2×#92687⋊C3, … |
| maximal quotient of | C3×ΓL(1,𝔽8)×#208 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 641);