#72222331

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

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×#104 ⋊ C₂×C₅
Hall	#10622435 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #453
Sylow	C₁₁ ⋊ C₅×#453

Subgroups

	subgroup	G/H
center	C₂	#66935⋊C₅
commutator, cosocle	C₂×C₄×C₁₁	C₂³×C₅
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₁₁)
Fitting	C₁₁×#104	C₂×C₅
socle	C₂×C₁₁	C₅×#120
hypercenter	#104	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#453
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

336 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,5,2^3,11|b^2=e,f^2=g,b^a=bf,c^a=cg,c^b=ce,e^a=eg,f^a=fg,f^c=fg,h^c=h^10,h^d=h^3

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#453×AGL(1,𝔽₁₁), C₂×#72222331
maximal normal subgroup of	#453×AGL(1,𝔽₁₁), C₂×#72222331
maximal quotient of	C₂×#72222331

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