#72276357

order	2200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×C₁₁ ⋊ C₂×C₅
Hall	C₅×C₁₁⋊C₅ ⋊ D₄
Sylow	C₁₁ ⋊ C₅×C₂²⋊D₅

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×C₅⋊AGL(1,𝔽₁₁)
commutator, socle, cosocle	C₂×C₅×C₁₁	C₂²×C₅
Fitting	C₂²×C₅×C₁₁	C₂×C₅
hypercenter	C₂²	C₅⋊AGL(1,𝔽₁₁)
hypocenter	C₅×C₁₁	C₅×D₄
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂ ◁ C₂²

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 220

Polycyclic presentation:

2^2,5,2,5,11|a^2=d,b^a=bd,e^a=e^4,f^a=f^10,f^c=f^9

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×C₂²⋊AGL(1,𝔽₁₁), D₄×C₅⋊AGL(1,𝔽₁₁), C₂²⋊D₅×AGL(1,𝔽₁₁)
maximal normal subgroup of	D₅×C₂²⋊AGL(1,𝔽₁₁), D₄×C₅⋊AGL(1,𝔽₁₁), C₂²⋊D₅×AGL(1,𝔽₁₁)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 5, 11]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 6, 3, F.6^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);