Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #6382⋊C3 |
| Hall | #6382 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #106 |
| Sylow | C7 ⋊ C3×#106 |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^2,3,2^3,7|a^2=e,b^2=d,e^2=f,b^a=bef,g^a=g^6,e^b=ef,g^c=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | #89533⋊C3, #89538⋊C3, #89549⋊C3, … |
| maximal normal subgroup of | #89533⋊C3, #89538⋊C3, #89549⋊C3, … |
| maximal quotient of | #89275⋊C3, #89282⋊C3, #89289⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 18);