#72228534

order	11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₅²×C₁₁ ⋊ C₂×C₄×C₅
Hall	C₅²×C₁₁⋊C₅ ⋊ C₂×C₄
Sylow	C₁₁ ⋊ C₅×D₅⋊AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₁	C₂×C₄×C₅
prime radical	#72228507	C₂
cosocle	C₁₁×C₅⋊D₅	C₂²×C₅
LCS	G ▷ C₅²×C₁₁
UCS	C₁

176 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 110

A minimal faithful permutation representation — 2 generators on 21 points:

(2,6,7,9,8,3,4,5,10,11)(12,13,15,14)(17,18,21,20);(1,2,5,4,8,10,9,11,6,7,3)(13,14,15,16)(18,19,21,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×D₅⋊AGL(1,𝔽₅)
maximal normal subgroup of	AGL(1,𝔽₁₁)×D₅⋊AGL(1,𝔽₅)

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