#72228524

order	2200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅²×C₁₁ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₂×Dic₁₁
Sylow	C₁₁ ⋊ D₅⋊AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₁	C₂×C₄
prime radical	D₅²×C₁₁	C₂
cosocle	C₁₁×C₅⋊D₅	C₂²
LCS	G ▷ C₅²×C₁₁
UCS	C₁

76 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228527, D₁₁×D₅⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×C₁₁⋊AGL(1,𝔽₅)
maximal normal subgroup of	#72228527, D₁₁×D₅⋊AGL(1,𝔽₅), AGL(1,𝔽₅)×C₁₁⋊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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 5, 11]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 4, 2, F.4^3);;
gap> SetConjugate(coll, 4, 3, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^3);;
gap> SetConjugate(coll, 5, 3, F.5^4);;
gap> SetConjugate(coll, 6, 2, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);