Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#6380 |
| Fitting | C3×C7×C22⋊C4 ⋊ C2 |
| Hall | C3×C7 ⋊ #99 |
| Sylow | C7 ⋊ C3×#99 |
104 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3)(4,5,7,9,10,8,6)(11,13)(12,15,16,17)(14,18);(5,6)(7,8)(9,10)(11,12)(13,17)(14,16)(15,18)123 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6380, #13052507, #13052509, … |
| maximal normal subgroup of | D3×#6380, #13052507, #13052509, … |
| maximal quotient of | C3×#89246, C3×#89247, C3×#89263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 145);