#72254540

order	7040
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₁₁×M₁₆ ⋊ C₂×C₅
Hall	#13079354 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #3184
Sylow	C₁₁ ⋊ C₅×#3184

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₄⋊AGL(1,𝔽₁₁)
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₁₁)
Fitting	C₄×C₁₁×M₁₆	C₂×C₅
socle	C₂²×C₁₁	C₂²×C₅×D₄
hypercenter	C₄×M₁₆	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#3184
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 704

Polycyclic presentation:

2^4,5,2^3,11|b^2=g,c^2=fh,d^2=g,f^2=h,c^a=cf,c^b=ch,d^a=dg,f^a=fh,i^a=i^10,i^e=i^9

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×#3184, #72254714, #72254756
maximal normal subgroup of	AGL(1,𝔽₁₁)×#3184, #72254714, #72254756

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