Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#6479 |
| Fitting | C3×C7×SD16 ⋊ C2 |
| Hall | C3×C7 ⋊ #135 |
| Sylow | C7 ⋊ C3×#135 |
124 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,7,9,10,8,6)(11,13,15,21)(12,17,16,19)(14,23,22,20)(18,25,24,26);(12,16)(13,20)(14,22)(17,25)(19,26)(21,23);(5,6)(7,8)(9,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,24)(23,26)105 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6479, #13059305, #13059306, … |
| maximal normal subgroup of | D3×#6479, #13059305, #13059306, … |
| maximal quotient of | C3×#89535, C3×#89559, C3×#89540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 809);