Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#6481 |
| Fitting | C3×C7×Q16 ⋊ C2 |
| Hall | C3×C7 ⋊ #137 |
| Sylow | C7 ⋊ C3×#137 |
120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 120 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)(11,13,15,19)(12,16,18,25)(14,23,24,21)(17,26,22,20);(11,12,15,18)(13,20,19,26)(14,22,24,17)(16,23,25,21);(5,6)(7,8)(9,10)(13,19)(16,25)(20,26)(21,23)96 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6481, #13059327, #13059329, … |
| maximal normal subgroup of | D3×#6481, #13059327, #13059329, … |
| maximal quotient of | C3×#89590, C3×#89563, C3×#89567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 811);