#72259160

order	7920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, Hall	C₃²×C₁₁ ⋊ C₅×SD₁₆
Hall	AGL(1,𝔽₉)⋊D₁₁ ⋊ C₅
Hall	C₃²×C₁₁⋊C₅ ⋊ SD₁₆
Hall	C₁₁⋊C₅ ⋊ AΓL(1,𝔽₉)
Sylow	C₁₁ ⋊ C₅×AΓL(1,𝔽₉)
Sylow	C₃² ⋊ #4314⋊C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₁₁×#157	C₂²×C₅
Fitting, socle, hypocenter	C₃²×C₁₁	C₅×SD₁₆
prime radical	#72259109	C₂
LCS	G ▷ C₁₁×#157 ▷ C₁₁×C₃⋊D₃ ▷ C₃²×C₁₁
UCS	C₁

100 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 99

Polycyclic presentation:

2^2,5,2^2,3^2,11|a^2=d,b^2=e,d^2=e,b^a=bde,d^b=de,f^a=f^2g^2,f^b=fg,f^d=f^2g,f^e=f^2,g^a=f^2,g^b=fg^2,g^d=fg,g^e=g^2,h^b=h^10,h^c=h^4

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×AΓL(1,𝔽₉)
maximal normal subgroup of	AGL(1,𝔽₁₁)×AΓL(1,𝔽₉)

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