Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #6435⋊C3 |
| Fitting | C42×C7 ⋊ C2×C3 |
| Hall | #6435 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #117 |
| Sylow | C7 ⋊ C3×#117 |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|b^2=f,c^2=e,c^a=cf,g^a=g^6,g^d=g^476 conjugacy classes — view the full character table.
| maximal subgroup of | #89461⋊C3, #89462⋊C3, #89480⋊C3, … |
| maximal normal subgroup of | #89461⋊C3, #89462⋊C3, #89480⋊C3, … |
| maximal quotient of | #89439⋊C3, #89422⋊C3, #89423⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 290);