Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6513⋊C3 |
| Fitting | C7×#41 ⋊ C2×C3 |
| Hall | #6513 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #135 |
| Sylow | C7 ⋊ C3×#135 |
124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|b^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^6,e^c=ef,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | #89595⋊C3, #89596⋊C3, #89916⋊C3, … |
| maximal normal subgroup of | #89595⋊C3, #89596⋊C3, #89916⋊C3, … |
| maximal quotient of | #89768⋊C3, #89770⋊C3, #89774⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 368);