#72214718

order	4840
automorphism	?
homology	Abelianization (H₁) = C₅×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow	C₁₁² ⋊ C₅×C₈
Hall	#11757261 ⋊ C₅
Hall	#10597243 ⋊ C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₁²	C₅×C₈
prime radical	#11925892	C₄
cosocle	#92150	C₂×C₅
LCS	G ▷ C₁₁²
UCS	C₁

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,5,2^2,11^2|a^2=c,c^2=d,e^a=e^9f^3,e^b=e^5,e^c=e^7f^2,e^d=e^10,f^a=ef^10,f^b=f^5,f^c=e^8f^4,f^d=f^10

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₂₁), #72224505
maximal normal subgroup of	AGL(1,𝔽₁₂₁), #72224505

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, 2, 11, 11]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^9*F.6^3);;
gap> SetConjugate(coll, 5, 2, F.5^5);;
gap> SetConjugate(coll, 5, 3, F.5^7*F.6^2);;
gap> SetConjugate(coll, 5, 4, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.5*F.6^10);;
gap> SetConjugate(coll, 6, 2, F.6^5);;
gap> SetConjugate(coll, 6, 3, F.5^8*F.6^4);;
gap> SetConjugate(coll, 6, 4, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);