#72217352

order	5280
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Hall	#11761014 ⋊ C₅
Hall	C₃×C₁₁⋊C₅ ⋊ #120
Hall	C₁₁⋊C₅ ⋊ #937
Hall	C₃×C₁₁ ⋊ C₅×#120
Sylow	C₁₁ ⋊ C₅×#937
Sylow	C₃ ⋊ #67006⋊C₅

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×AGL(1,𝔽₁₁)
commutator, socle, cosocle	C₂²×C₃×C₁₁	C₂³×C₅
Fitting	C₂³×C₃×C₁₁	C₂²×C₅
hypercenter	C₂³	D₃×AGL(1,𝔽₁₁)
hypocenter	C₃×C₁₁	C₅×#120
LCS	G ▷ C₂²×C₃×C₁₁ ▷ C₃×C₁₁
UCS	C₁ ◁ C₂² ◁ C₂³

520 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 97 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|c^2=e,c^a=ce,c^b=cef,g^b=g^2,h^c=h^10,h^d=h^9

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#67006⋊C₅, #72217599, #937×AGL(1,𝔽₁₁)
maximal normal subgroup of	D₃×#67006⋊C₅, #72217599, #937×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, 5, 2, 2, 3, 11]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.8^10);;
gap> SetConjugate(coll, 8, 4, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);