#72209869

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11715690
commutator, cosocle, hypocenter	#2273⋊C₃	C₂²
Frattini, socle	C₂³	C₂×#1020
Fitting	#35390	D₃
prime radical	#35390⋊C₃	C₂
LCS	G ▷ #2273⋊C₃
UCS	C₁ ◁ C₂

1255 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 1255 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^7|b^2=j,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=bi,c^a=ch,c^b=c^2,d^a=dh,d^b=ei,d^c=eh,e^a=ehi,e^b=di,e^c=de,f^a=fhi,f^b=fgi,f^c=ghi,f^d=fj,g^a=gi,g^b=gi,g^c=fg,g^d=gj,g^e=gj,h^b=hi,h^c=hi,i^c=h

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341125
maximal normal subgroup of	#72341125
maximal quotient of	#72209870

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