Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6482⋊C3 |
| Fitting | C7×Q16 ⋊ C2×C3 |
| Hall | #6482 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #135 |
| Sylow | C7 ⋊ C3×#135 |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|b^2=f,c^2=f,e^2=f,b^a=bf,c^a=cf,g^a=g^6,c^b=ce,e^b=ef,e^c=ef,g^d=g^449 conjugacy classes — view the full character table.
| maximal subgroup of | #89686⋊C3, #89688⋊C3, #89690⋊C3, … |
| maximal normal subgroup of | #89686⋊C3, #89688⋊C3, #89690⋊C3, … |
| maximal quotient of | #89591⋊C3, #89568⋊C3, #89572⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 337);