#72259148

order	3960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, Hall	C₃²×C₁₁ ⋊ C₅×D₄
Hall	D₃²⋊D₁₁ ⋊ C₅
Hall	C₃²×C₁₁⋊C₅ ⋊ D₄
Hall	C₁₁⋊C₅ ⋊ GO⁺(4,𝔽₂)
Sylow	C₁₁ ⋊ C₅×GO⁺(4,𝔽₂)
Sylow	C₃² ⋊ C₂²⋊AGL(1,𝔽₁₁)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₁₁×C₃⋊D₃	C₂²×C₅
Fitting, socle, hypocenter	C₃²×C₁₁	C₅×D₄
LCS	G ▷ C₁₁×C₃⋊D₃ ▷ C₃²×C₁₁
UCS	C₁

104 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 66

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	GO⁺(4,𝔽₂)×AGL(1,𝔽₁₁)
maximal normal subgroup of	GO⁺(4,𝔽₂)×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, 2, 5, 2, 3, 3, 11]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 4, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^10);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);