Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#6502 |
| Fitting | C2×C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ #121 |
| Sylow | C7 ⋊ C3×#121 |
188 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 188 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) = 18 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(4,5,7,9,10,8,6)(12,14)(16,18);(15,16)(17,18);(5,6)(7,8)(9,10)(11,12)(13,14)(16,18)132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6502, #13059928, #13059930, … |
| maximal normal subgroup of | D3×#6502, #13059928, #13059930, … |
| maximal quotient of | C3×#89714, C3×#89729, C3×#89717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 832);