#72248917

order	3520
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₈×C₁₁ ⋊ C₂×C₅
Hall	#10622308 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ #493
Sylow	C₁₁ ⋊ C₅×#493

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₁₁)
commutator, cosocle	C₂×C₄×C₁₁	C₂³×C₅
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₁₁)
Fitting	C₄×C₈×C₁₁	C₂×C₅
socle	C₂²×C₁₁	C₂×C₅×D₄
hypercenter	C₄×C₈	AGL(1,𝔽₁₁)
hypocenter	C₁₁	C₅×#493
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 352

Polycyclic presentation:

2^3,5,2^3,11|b^2=eg,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^10,h^d=h^9

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#493×AGL(1,𝔽₁₁), #72248969, #72248979, …
maximal normal subgroup of	#493×AGL(1,𝔽₁₁), #72248969, #72248979, …
maximal quotient of	#72249013, #72254494

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