Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#6509 |
| Fitting | C2×C3×C7×Q8 ⋊ C2 |
| Hall | C3×C7 ⋊ C22⋊Q8 |
| Sylow | C7 ⋊ C3×C22⋊Q8 |
148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,7,9,10,8,6)(12,14)(15,17,18,22)(16,19,20,21);(15,16,18,20)(17,21,22,19);(5,6)(7,8)(9,10)(11,12)(13,14)132 conjugacy classes — view the full character table.
| maximal subgroup of | #13054567, #13054568, #13054578, … |
| maximal normal subgroup of | #13054567, #13054568, #13054578, … |
| maximal quotient of | C3×#89750, C3×#89745, C3×#89756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 839);