#72248757

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×M₃₂ ⋊ C₂×C₅
Hall	#10622521 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #509
Sylow	C₁₁ ⋊ C₅×#509

Subgroups

	subgroup	G/H
center	C₂	C₂×D₈₈⋊C₅
commutator, cosocle	C₈×C₁₁	C₂³×C₅
Frattini	C₈	C₂²×AGL(1,𝔽₁₁)
Fitting	C₁₁×M₃₂	C₂×C₅
socle	C₂×C₁₁	C₂×C₅×D₈
hypercenter	M₃₂	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#509
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

180 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 176

Polycyclic presentation:

2^3,5,2^3,11|a^2=e,e^2=fg,f^2=g,b^a=be,c^a=cg,e^b=ef,f^b=fg,h^b=h^10,h^d=h^9

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#509×AGL(1,𝔽₁₁), #72248784, #72248786
maximal normal subgroup of	#509×AGL(1,𝔽₁₁), #72248784, #72248786

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