Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #6499⋊C3 |
| Hall | #6499 ⋊ 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|a^2=f,b^2=e,c^a=cef,g^a=g^6,c^b=ce,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | #89268⋊C3, #89518⋊C3, #89559⋊C3, … |
| maximal normal subgroup of | #89268⋊C3, #89518⋊C3, #89559⋊C3, … |
| maximal quotient of | #89719⋊C3, #89725⋊C3, #89717⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 354);