#72269559

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×#131 ⋊ C₂×C₅
Hall	#10623283 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #576
Sylow	C₁₁ ⋊ C₅×#576

Subgroups

	subgroup	G/H
center	C₂	C₂²×C₄⋊AGL(1,𝔽₁₁)
commutator, cosocle	C₄×C₁₁	C₂⁴×C₅
Frattini	C₄	C₂³×AGL(1,𝔽₁₁)
Fitting	C₁₁×#131	C₂×C₅
socle	C₂×C₁₁	C₂²×C₅×D₄
hypercenter	#131	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#576
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

536 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 176

Polycyclic presentation:

2^4,5,2^2,11|a^2=f,f^2=g,b^a=bfg,c^a=cg,d^a=dg,d^c=dg,f^b=fg,h^b=h^10,h^e=h^9

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269564, #72269596, #576×AGL(1,𝔽₁₁)
maximal normal subgroup of	#72269564, #72269596, #576×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, 2, 11]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^10);;
gap> SetConjugate(coll, 8, 5, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);