#72221925

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72221849 ⋊ C₂
derived	#14554854 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#14554854	C₂×C₃
Frattini	C₃³	C₃×C₃⋊(C₃⋊(C₃⋊D₃))
Fitting	#72221849	C₂
socle	C₃	C₃×#13091502
LCS	G ▷ #14554854
UCS	C₁

11235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^8|c^a=c^2,d^a=d^2,d^c=dh,e^a=e^2,e^b=ei,e^c=eg,f^a=f^2,f^d=fg,f^e=fg^2,g^a=gi^2,g^c=gi,g^f=gi,h^f=hi^2,i^a=i^2

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222000, #72243867
maximal normal subgroup of	#72222000, #72243867

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);