#72228501

order	5500
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₅²×C₁₁ ⋊ C₄×C₅
Hall	C₅²×C₁₁⋊C₅ ⋊ C₄
Sylow	C₁₁ ⋊ C₅×#1043

Subgroups

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

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 275

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×AGL(1,𝔽₁₁), #72228527, #72228534
maximal normal subgroup of	#1043×AGL(1,𝔽₁₁), #72228527, #72228534

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