#72248965

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	#10623329 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #518
Sylow	C₁₁ ⋊ C₅×#518

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₁₁)
commutator	C₂×C₁₁	C₂³×C₄×C₅
Fitting	C₂²×D₄×C₁₁	C₂×C₅
socle, cosocle	C₂²×C₁₁	C₂⁴×C₅
prime radical	C₂²×D₄×C₁₁⋊C₅	C₂
hypercenter	C₂²×D₄	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#518
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

676 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 312 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,c^2=g,c^b=cg,d^a=dg,h^a=h^10,h^e=h^9

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248978, #72254537, #518×AGL(1,𝔽₁₁)
maximal normal subgroup of	#72248978, #72254537, #518×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, 3, F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 8, 5, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);