Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#6472 |
| Fitting | C3×C7×M16 ⋊ C2 |
| Hall | C3×C7 ⋊ #137 |
| Sylow | C7 ⋊ C3×#137 |
120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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)(11,14,15,22,16,23,13,20)(12,18,17,21,19,26,24,25);(11,13,16,15)(12,17,19,24)(14,22,23,20)(18,25,26,21);(5,6)(7,8)(9,10)(11,12)(13,17)(14,21)(15,24)(16,19)(18,22)(20,26)(23,25)123 conjugacy classes — view the full character table.
| maximal subgroup of | #13053589, #13053591, #13053595, … |
| maximal normal subgroup of | #13053589, #13053591, #13053595, … |
| maximal quotient of | C3×#89474, C3×#89486, C3×#89475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 802);