#72276402

order	4400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₅×C₁₁⋊C₅ ⋊ C₂²⋊C₄
Sylow	C₁₁ ⋊ C₅×C₂²⋊AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₅)×AGL(1,𝔽₁₁)
commutator, Fitting, socle	C₂×C₅×C₁₁	C₂×C₄×C₅
prime radical	C₂×D₅×AGL(1,𝔽₁₁)	C₂
cosocle	C₂×D₅×C₁₁	C₂²×C₅
hypocenter	C₅×C₁₁	C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂

146 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 220

Polycyclic presentation:

2^2,5,2^2,5,11|a^2=d,b^a=be,f^a=f^3,f^d=f^4,g^b=g^10,g^c=g^9

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊AGL(1,𝔽₅)×AGL(1,𝔽₁₁), AGL(1,𝔽₅)×C₂²⋊AGL(1,𝔽₁₁), #72276412, …
maximal normal subgroup of	C₂²⋊AGL(1,𝔽₅)×AGL(1,𝔽₁₁), AGL(1,𝔽₅)×C₂²⋊AGL(1,𝔽₁₁), #72276412, …

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