Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6479⋊C3 |
| Fitting | C7×SD16 ⋊ C2×C3 |
| Hall | #6479 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #135 |
| Sylow | C7 ⋊ C3×#135 |
124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|c^2=f,e^2=f,b^a=bf,c^a=cf,g^a=g^6,c^b=ce,e^b=ef,e^c=ef,g^d=g^449 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6479⋊C3, #90453⋊C3, #135×AGL(1,𝔽7), … |
| maximal normal subgroup of | C2×#6479⋊C3, #90453⋊C3, #135×AGL(1,𝔽7), … |
| maximal quotient of | #89535⋊C3, #89559⋊C3, #89540⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 334);