Finite Group DatabasebetaFinite Group Databasebeta| order | 13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C11 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | 11−1+2 ⋊ C2×C5 |
| derived | C121 ⋊ C2×C5×C11 |
| Hall | #72230790 ⋊ C2 |
| Hall | #72230789 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C121 | C2×C5×C11 |
| Frattini, socle | C11 | C11×AGL(1,𝔽11) |
| Fitting | 11−1+2 | C2×C5 |
| LCS | G ▷ C121 | |
| UCS | C1 | |
25 conjugacy classes of subgroups.
| transitive degree | μt(G) = 121 |
Polycyclic presentation:
2,5,11^3|d^11=e,d^a=d^10e^10,d^b=d^4e^7,d^c=de^10,e^a=e^10,e^b=e^4122 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72230792 |
| maximal normal subgroup of | C2×#72230792 |
| maximal quotient of | C2×#72230792 |
| automorphism group of | D121 |
| holomorph of | C121 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 5, 11, 11, 11]);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 4, 1, F.4^10*F.5^10);;
gap> SetConjugate(coll, 4, 2, F.4^4*F.5^7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5^10);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);