Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#911 |
| Fitting | C3×C7×D8 ⋊ C2 |
| Hall | C3×C7 ⋊ AGL(1,ℤ/8) |
| Sylow | C3 ⋊ C7×AGL(1,ℤ/8) |
136 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 136 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) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3,4,5,6,7)(8,9,10)(11,12)(13,18)(14,15)(16,17);(12,16)(13,17)(15,18);(9,10)(12,15)(16,18)126 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#911, #13059275, #13059277, … |
| maximal normal subgroup of | D7×#911, #13059275, #13059277, … |
| maximal quotient of | C7×#4761, C7×#4762, C7×#4763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 890);