Finite Group DatabasebetaFinite Group Databasebeta| order | 7040 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| Fitting | C11×#493 ⋊ C2×C5 |
| Hall | #13068230 ⋊ C5 |
| Hall | C11⋊C5 ⋊ #1712 |
| Sylow | C11 ⋊ C5×#1712 |
560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 704 |
Polycyclic presentation:
2^3,5,2^4,11|c^2=fh,e^2=g,f^2=h,b^a=be,c^a=cf,c^b=cf,e^a=eg,e^b=eg,f^a=fh,f^b=fh,i^b=i^10,i^d=i^9212 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽11)×#1712, #72254673, #72254702, … |
| maximal normal subgroup of | AGL(1,𝔽11)×#1712, #72254673, #72254702, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 2, 2, 2, 2, 11]);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^10);;
gap> SetConjugate(coll, 9, 4, F.9^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);