Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #6374⋊C3 |
| Fitting | C7×C16 ⋊ C2×C3 |
| Hall | #6374 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ SD32 |
| Sylow | C7 ⋊ C3×SD32 |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,2^3,7|b^2=de,d^2=e,e^2=f,b^a=bd,d^a=def,e^a=ef,g^a=g^6,g^c=g^449 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6374⋊C3, #89676⋊C3, #89680⋊C3, … |
| maximal normal subgroup of | C2×#6374⋊C3, #89676⋊C3, #89680⋊C3, … |
| maximal quotient of | #89300⋊C3, #89304⋊C3, #89298⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 10);