Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#916 |
| Fitting | C3×C7×SD16 ⋊ C2 |
| Hall | C3×C7 ⋊ #135 |
| Sylow | C3 ⋊ C7×#135 |
124 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 124 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) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5,6,7)(8,9,10)(11,13,15,21)(12,17,16,19)(14,23,22,20)(18,25,24,26);(12,16)(13,20)(14,22)(17,25)(19,26)(21,23);(9,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,24)(23,26)147 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#916, #13059301, #13059303, … |
| maximal normal subgroup of | D7×#916, #13059301, #13059303, … |
| maximal quotient of | C7×#4763, C7×#4787, C7×#4768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 895);