#72249016

order	3520
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Hall	#10622093 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #322
Sylow	C₁₁ ⋊ C₅×#322

Subgroups

	subgroup	G/H
center, Frattini	C₄²	C₂×AGL(1,𝔽₁₁)
commutator	C₂×C₁₁	C₄×C₅×C₈
Fitting	C₄×C₈×C₁₁	C₂×C₅
socle	C₂²×C₁₁	C₄²×C₅
prime radical	C₂²×C₁₁⋊C₅	C₄²
cosocle	C₄²×C₁₁	C₂²×C₅
hypercenter	C₄×C₈	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#322
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄² ◁ C₄×C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 704

Polycyclic presentation:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#322×AGL(1,𝔽₁₁), #72249029, #72249031, …
maximal normal subgroup of	#322×AGL(1,𝔽₁₁), #72249029, #72249031, …

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