Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6481⋊C3 |
| Fitting | C7×Q16 ⋊ C2×C3 |
| Hall | #6481 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #137 |
| Sylow | C7 ⋊ C3×#137 |
120 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 47 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=f,e^2=f,c^a=cf,g^a=g^6,c^b=ce,e^b=ef,e^c=ef,g^d=g^440 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6481⋊C3, #90457⋊C3, #90459⋊C3, … |
| maximal normal subgroup of | C2×#6481⋊C3, #90457⋊C3, #90459⋊C3, … |
| maximal quotient of | #89590⋊C3, #89563⋊C3, #89567⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 336);