Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6446⋊C3 |
| Fitting | C7×C22⋊C4 ⋊ C2×C3 |
| Hall | #6446 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #123 |
| Sylow | C7 ⋊ C3×#123 |
156 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|b^2=e,c^a=ce,g^a=g^6,c^b=cf,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6446⋊C3, #90181⋊C3, #90185⋊C3, … |
| maximal normal subgroup of | C2×#6446⋊C3, #90181⋊C3, #90185⋊C3, … |
| maximal quotient of | #89424⋊C3, #89429⋊C3, #89430⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 301);