Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #6510⋊C3 |
| Fitting | C2×C7×Q8 ⋊ C2×C3 |
| Hall | #6510 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #124 |
| Sylow | C7 ⋊ C3×#124 |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|b^2=f,c^2=f,b^a=bf,c^a=ce,g^a=g^6,c^b=cf,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | #89272⋊C3, #89525⋊C3, #89591⋊C3, … |
| maximal normal subgroup of | #89272⋊C3, #89525⋊C3, #89591⋊C3, … |
| maximal quotient of | #89749⋊C3, #89758⋊C3, #89760⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 365);