Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6441⋊C3 |
| Hall | #6441 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C22⋊Q8 |
| Sylow | C7 ⋊ C3×C22⋊Q8 |
148 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 59 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|a^2=e,c^2=e,c^a=ce,g^a=g^6,c^b=cf,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6441⋊C3, #90174⋊C3, #90181⋊C3, … |
| maximal normal subgroup of | C2×#6441⋊C3, #90174⋊C3, #90181⋊C3, … |
| maximal quotient of | #89418⋊C3, #89419⋊C3, #89428⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 296);