#72209790

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting

#10935 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11711026
commutator, cosocle, hypocenter	#2273⋊C₃	C₂²
Frattini	C₂⁵	C₂×S₄
Fitting	#10935	D₃
socle	C₂³	#1020⋊C₂
LCS	G ▷ #2273⋊C₃
UCS	C₁ ◁ C₂

779 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 16 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=ij,d^2=i,e^2=h,b^a=bi,c^a=chj,c^b=c^2hi,d^a=dfj,d^b=df,d^c=eij,e^a=efgj,e^b=deghij,e^c=dej,e^d=ej,f^c=fg,f^e=fj,g^b=fg,g^c=f,g^d=gj,g^e=gj,h^b=hij,h^c=hij,i^c=h

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209792
maximal normal subgroup of	#72209792
maximal quotient of	#72227714

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