Finite Group DatabasebetaFinite Group Databasebeta| order | 2640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #10587141 ⋊ C5 |
| Hall | C3×C11⋊C5 ⋊ D8 |
| Hall | C11⋊C5 ⋊ #213 |
| Hall | C3×C11 ⋊ C5×D8 |
| Sylow | C11 ⋊ C5×#213 |
| Sylow | C3 ⋊ #4323⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #63140⋊C5 |
| commutator, Fitting, cosocle | C3×C4×C11 | C22×C5 |
| Frattini, hypercenter | C4 | D3×AGL(1,𝔽11) |
| socle | C2×C3×C11 | C5×D4 |
| hypocenter | C3×C11 | C5×D8 |
| LCS | G ▷ C3×C4×C11 ▷ C2×C3×C11 ▷ C3×C11 | |
| UCS | C1 ◁ C2 ◁ C4 | |
88 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,5,2^2,3,11|a^2=d,d^2=e,b^a=bd,d^b=de,f^a=f^2,g^a=g^10,g^b=g^10,g^c=g^969 conjugacy classes — view the full character table.
| maximal subgroup of | #72216928, #72216938, #72216947, … |
| maximal normal subgroup of | #72216928, #72216938, #72216947, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 3, 11]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 7, 2, F.7^10);;
gap> SetConjugate(coll, 7, 3, F.7^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);