Finite Group DatabasebetaFinite Group Databasebeta| order | 3520 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C11×2−1+4 ⋊ C2×C5 |
| Hall | #10623457 ⋊ C5 |
| Hall | C11⋊C5 ⋊ #585 |
| Sylow | C11 ⋊ C5×#585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×AGL(1,𝔽11) |
| commutator, socle, cosocle | C2×C11 | C25×C5 |
| Fitting | C11×2−1+4 | C2×C5 |
| hypercenter | 2−1+4 | AGL(1,𝔽11) |
| hypocenter | C11 | C5×#585 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
1620 conjugacy classes of subgroups; only the 818 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 818 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:
2^5,5,2,11|a^2=g,e^2=g,c^a=cg,d^a=dg,d^b=dg,e^b=eg,h^c=h^10,h^f=h^4187 conjugacy classes — view the full character table.
| maximal subgroup of | #585×AGL(1,𝔽11), #72269596 |
| maximal normal subgroup of | #585×AGL(1,𝔽11), #72269596 |
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, 2, 5, 2, 11]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^10);;
gap> SetConjugate(coll, 8, 6, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);