Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6380⋊C3 |
| Fitting | C7×C22⋊C4 ⋊ C2×C3 |
| Hall | #6380 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #99 |
| Sylow | C7 ⋊ C3×#99 |
104 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^2,3,2^3,7|b^2=d,b^a=be,d^a=df,g^a=g^6,e^b=ef,g^c=g^443 conjugacy classes — view the full character table.
| maximal subgroup of | #89512⋊C3, #89513⋊C3, #89514⋊C3, … |
| maximal normal subgroup of | #89512⋊C3, #89513⋊C3, #89514⋊C3, … |
| maximal quotient of | #89246⋊C3, #89247⋊C3, #89263⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 16);