#72259113

order	3960
automorphism	?
homology	Abelianization (H₁) = C₅×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃²×C₁₁ ⋊ C₅×C₈
Hall	C₁₁⋊AGL(1,𝔽₉) ⋊ C₅
Hall	C₃²×C₁₁⋊C₅ ⋊ C₈
Hall	C₁₁⋊C₅ ⋊ AGL(1,𝔽₉)
Sylow	C₁₁ ⋊ C₅×AGL(1,𝔽₉)
Sylow	C₃² ⋊ (C₁₁⋊C₅)⋊C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₁₁	C₅×C₈
prime radical	C₃⋊D₃×C₁₁⋊C₅	C₄
cosocle	C₁₁×#157	C₂×C₅
LCS	G ▷ C₃²×C₁₁
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 99

Polycyclic presentation:

2,5,2^2,3^2,11|a^2=c,c^2=d,e^a=e^2f^2,e^c=e^2f,e^d=e^2,f^a=e^2,f^c=ef,f^d=f^2,g^a=g^10,g^b=g^4

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259115, #72259151, AGL(1,𝔽₉)×AGL(1,𝔽₁₁)
maximal normal subgroup of	#72259115, #72259151, AGL(1,𝔽₉)×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");;
gap> coll := SingleCollector(F, [2, 5, 2, 2, 3, 3, 11]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 3, F.5^2*F.6);;
gap> SetConjugate(coll, 5, 4, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 7, 2, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);