#72269573

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₁₁×H(5,𝔽₂) ⋊ C₂×C₅
Hall	#13083396 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #3585
Sylow	C₁₁ ⋊ C₅×#3585

Subgroups

	subgroup	G/H
center	C₂	C₂³×C₂²⋊AGL(1,𝔽₁₁)
commutator, cosocle	C₄×C₁₁	C₂⁵×C₅
Frattini	C₄	C₂⁴×AGL(1,𝔽₁₁)
Fitting	C₂×C₁₁×H(5,𝔽₂)	C₂×C₅
socle	C₂×C₁₁	C₂³×C₅×D₄
hypercenter	C₂×H(5,𝔽₂)	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#3585
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

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=g,g^2=h,b^a=bgh,d^a=dh,e^a=eh,e^c=eh,g^b=gh,i^a=i^10,i^f=i^9

Conjugacy classes & character table

239 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.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*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, 3, F.5*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> SetConjugate(coll, 9, 6, F.9^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);