Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#894 |
| Fitting | C3×C7×C4⋊C4 ⋊ C2 |
| Hall | C3×C7 ⋊ #123 |
| Sylow | C3 ⋊ C7×#123 |
156 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5,6,7)(8,9,10)(11,12,13,14)(16,20)(19,22);(12,14)(15,17,18,21)(16,19,20,22);(9,10)(15,16)(17,19)(18,20)(21,22)168 conjugacy classes — view the full character table.
| maximal subgroup of | #13058564, #13058566, #13058568, … |
| maximal normal subgroup of | #13058564, #13058566, #13058568, … |
| maximal quotient of | C7×#4660, C7×#4655, C7×#4664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 873);