Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#951 |
| Hall | C3×C7 ⋊ #137 |
| Sylow | C3 ⋊ C7×#137 |
120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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)(12,17)(13,20)(14,22)(16,23)(18,19)(24,25);(11,13)(12,16)(14,21)(15,20)(17,25)(18,23)(19,24)(22,26);(9,10)(11,12,15,19)(13,16,20,24)(14,17,22,18)(21,25,26,23)147 conjugacy classes — view the full character table.
| maximal subgroup of | #13054803, #13054823, #13054826, … |
| maximal normal subgroup of | #13054803, #13054823, #13054826, … |
| maximal quotient of | C7×#4996, C7×#4995, C7×#5004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 930);