#72217005

order	5280
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₃×Q₈×C₁₁ ⋊ C₂²×C₅
Hall	#11760956 ⋊ C₅
Hall	C₃×C₁₁⋊C₅ ⋊ AGL(1,ℤ/8)
Hall	C₁₁⋊C₅ ⋊ #914
Hall	C₃×C₁₁ ⋊ C₅×AGL(1,ℤ/8)
Sylow	C₁₁ ⋊ C₅×#914
Sylow	C₃ ⋊ #66967⋊C₅

Subgroups

	subgroup	G/H
center	C₂	#72216761
commutator, cosocle	C₃×C₄×C₁₁	C₂³×C₅
Frattini	C₄	C₂×D₃×AGL(1,𝔽₁₁)
Fitting	C₃×Q₈×C₁₁	C₂²×C₅
socle	C₂×C₃×C₁₁	C₂×C₅×D₄
hypercenter	Q₈	D₃×AGL(1,𝔽₁₁)
hypocenter	C₃×C₁₁	C₅×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₁₁ ▷ C₂×C₃×C₁₁ ▷ C₃×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

272 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 65 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:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#66967⋊C₅, #914×AGL(1,𝔽₁₁), #72217438
maximal normal subgroup of	D₃×#66967⋊C₅, #914×AGL(1,𝔽₁₁), #72217438

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