#72222903

order	29403
automorphism	?
homology	Abelianization (H₁) = C₁₂₁ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₃⁵ ⋊ C₁₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃⁵	C₁₂₁
prime radical, cosocle	C₃⁵⋊C₁₁	C₁₁
LCS	G ▷ C₃⁵
UCS	C₁

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

11^2,3^5|a^11=b,c^a=cd,c^b=cd^2e,d^a=d^2e^2f,d^b=cde^2f,e^a=cde^2f^2,e^b=c^2dfg^2,f^a=df^2g,f^b=d^2e^2,g^a=c^2d^2e,g^b=cde^2f^2

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₂₄₃)
maximal normal subgroup of	AGL(1,𝔽₂₄₃)

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