#72269648

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

Aliases

No aliases for this group.

Splittings

Fitting	C₁₁×#108 ⋊ C₂×C₅
Hall	#10622506 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #430
Sylow	C₁₁ ⋊ C₅×#430

Subgroups

	subgroup	G/H
center	C₄	D₄×AGL(1,𝔽₁₁)
commutator	C₄×C₁₁	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₁₁)
Fitting	C₁₁×#108	C₂×C₅
socle	C₂×C₁₁	C₂×C₅×C₄⋊C₄
prime radical	C₂²×AGL(1,𝔽₁₁)	Q₈
cosocle	C₂×C₄×C₁₁	C₂³×C₅
hypercenter	#108	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#430
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

204 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 90 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:

2^3,5,2^3,11|a^2=f,b^2=fg,e^2=g,f^2=g,b^a=be,c^a=cg,e^a=eg,e^b=eg,h^c=h^10,h^d=h^4

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

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