Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | SD16×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×SD16) |
| unambiguous semidirect product | (D7×SD16)⋊C3 |
| Fitting | C7×SD16 ⋊ C2×C3 |
| Hall | D7×SD16 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×SD16 |
| Sylow | C7 ⋊ C2×C3×SD16 |
136 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD16×AGL(1,𝔽7), #90453⋊C3, #135×AGL(1,𝔽7), … |
| maximal normal subgroup of | C2×SD16×AGL(1,𝔽7), #90453⋊C3, #135×AGL(1,𝔽7), … |
| maximal quotient of | #89532⋊C3, #89533⋊C3, #89534⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 331);