Finite Group DatabasebetaFinite Group Databasebeta| order | 3520 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #10622204 ⋊ C5 |
| Hall | C11⋊C5 ⋊ #360 |
| Sylow | C11 ⋊ C5×#360 |
116 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 176 |
Polycyclic presentation:
2^2,5,2^4,11|a^2=d,b^2=def,d^2=f,e^2=f,f^2=g,b^a=be,d^b=df,e^a=eg,e^b=ef,f^b=fg,h^a=h^10,h^b=h^10,h^c=h^994 conjugacy classes — view the full character table.
| maximal subgroup of | #360×AGL(1,𝔽11), #72248787, #72248792 |
| maximal normal subgroup of | #360×AGL(1,𝔽11), #72248787, #72248792 |
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");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 2, 2, 11]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.5*F.6);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 8, 2, F.8^10);;
gap> SetConjugate(coll, 8, 3, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);