#72248715

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄×D₄×C₁₁ ⋊ C₂×C₅
Hall	#10622627 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #442
Sylow	C₁₁ ⋊ C₅×#442

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊AGL(1,𝔽₁₁)
commutator	C₄×C₁₁	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₁₁)
Fitting	C₄×D₄×C₁₁	C₂×C₅
socle	C₂²×C₁₁	C₂×C₅×D₄
prime radical	C₂×#4323⋊C₅	C₂
cosocle	C₂×C₄×C₁₁	C₂³×C₅
hypercenter	C₄×D₄	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#442
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 352

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#442×AGL(1,𝔽₁₁), #72254550, #72254557
maximal normal subgroup of	#442×AGL(1,𝔽₁₁), #72254550, #72254557
maximal quotient of	#72248789, #72248792

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