Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #6456⋊C3 |
| Fitting | C7×C4⋊C4 ⋊ C2×C3 |
| Hall | #6456 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C4×D4 |
| Sylow | C7 ⋊ C3×C4×D4 |
188 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 75 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=f,c^2=e,b^a=bf,g^a=g^6,c^b=cf,g^d=g^470 conjugacy classes — view the full character table.
| maximal subgroup of | #89558⋊C3, #89559⋊C3, #89557⋊C3, … |
| maximal normal subgroup of | #89558⋊C3, #89559⋊C3, #89557⋊C3, … |
| maximal quotient of | #89438⋊C3, #89426⋊C3, #89437⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 311);