#72269587

order	7040
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₁₁×AGL(1,ℤ/8) ⋊ C₂×C₅
Hall	#13083392 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #3585
Sylow	C₁₁ ⋊ C₅×#3585

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×AGL(1,𝔽₁₁)
commutator, cosocle	C₄×C₁₁	C₂⁵×C₅
Frattini	C₄	C₂⁴×AGL(1,𝔽₁₁)
Fitting	C₂×C₁₁×AGL(1,ℤ/8)	C₂×C₅
socle	C₂×C₁₁	C₂³×C₅×D₄
hypercenter	C₂×AGL(1,ℤ/8)	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#3585
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

2928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 176

Polycyclic presentation:

2^5,5,2^2,11|a^2=h,g^2=h,b^a=bg,c^b=cgh,d^a=dh,e^a=eh,e^b=eh,g^a=gh,g^b=gh,g^c=gh,i^a=i^10,i^f=i^4

Conjugacy classes & character table

227 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×#3585
maximal normal subgroup of	AGL(1,𝔽₁₁)×#3585

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", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 2, 2, 11]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> SetConjugate(coll, 9, 6, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);