Finite Group DatabasebetaFinite Group Databasebeta| order | 3520 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C11×#120 ⋊ C2×C5 |
| Hall | #10623121 ⋊ C5 |
| Hall | C11⋊C5 ⋊ D42 |
| Sylow | C11 ⋊ C5×D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽11) |
| commutator, socle, cosocle | C22×C11 | C24×C5 |
| Fitting | C11×#120 | C2×C5 |
| hypercenter | #120 | AGL(1,𝔽11) |
| hypocenter | C11 | C5×D42 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ #120 | |
856 conjugacy classes of subgroups.
| transitive degree | μt(G) = 176 |
Polycyclic presentation:
2^4,5,2^2,11|b^a=bf,c^a=cf,d^c=dg,h^b=h^10,h^e=h^9139 conjugacy classes — view the full character table.
| maximal subgroup of | #72248973, D42×AGL(1,𝔽11), #72254515, … |
| maximal normal subgroup of | #72248973, D42×AGL(1,𝔽11), #72254515, … |
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, 2, 2, 5, 2, 2, 11]);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^10);;
gap> SetConjugate(coll, 8, 5, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);