#72217140

order	5280
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×Q₈×C₁₁ ⋊ C₂²×C₅
Hall	#11761402 ⋊ C₅
Hall	C₃×C₁₁⋊C₅ ⋊ C₂×#41
Hall	C₁₁⋊C₅ ⋊ D₃×#41
Hall	C₃×C₁₁ ⋊ C₂×C₅×#41
Sylow	C₁₁ ⋊ C₅×D₃×#41
Sylow	C₃ ⋊ #41×AGL(1,𝔽₁₁)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×AGL(1,𝔽₁₁)
commutator, socle, cosocle	C₂×C₃×C₁₁	C₂⁴×C₅
Fitting	C₃×Q₈×C₁₁	C₂²×C₅
hypercenter	Q₈	D₃×AGL(1,𝔽₁₁)
hypocenter	C₃×C₁₁	C₂×C₅×#41
LCS	G ▷ C₂×C₃×C₁₁ ▷ C₃×C₁₁
UCS	C₁ ◁ C₂ ◁ Q₈

656 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 197 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:

2^4,5,2,3,11|d^2=f,b^a=bf,d^a=df,g^c=g^2,h^a=h^10,h^b=h^10,h^c=h^10,h^e=h^9

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217367, D₃×#41×AGL(1,𝔽₁₁)
maximal normal subgroup of	#72217367, D₃×#41×AGL(1,𝔽₁₁)

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