Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #6433⋊C3 |
| Hall | #6433 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #125 |
| Sylow | C7 ⋊ C3×#125 |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,2^2,7|a^2=f,b^2=e,c^2=f,b^a=bf,c^a=ce,g^a=g^6,g^d=g^458 conjugacy classes — view the full character table.
| maximal subgroup of | #89260⋊C3, #89455⋊C3, #89469⋊C3, … |
| maximal normal subgroup of | #89260⋊C3, #89455⋊C3, #89469⋊C3, … |
| maximal quotient of | #89429⋊C3, #89432⋊C3, #89705⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 288);