#72213204

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

Aliases

No aliases for this group.

Splittings

Fitting

#10304 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11711452
commutator, cosocle, hypocenter	#1273⋊C₃	C₂²
Frattini	C₂×C₄²	C₂×S₄
Fitting	#10304	D₃
socle	C₂³	#857⋊C₂
LCS	G ▷ #1273⋊C₃
UCS	C₁ ◁ C₂

331 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 14 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|a^2=hj,b^2=h,d^2=f,e^2=g,f^2=i,g^2=j,b^a=bj,c^a=c^2i,c^b=cij,d^a=defhij,d^b=dfhi,d^c=efhij,e^a=eh,e^b=eghj,e^c=deh,e^d=eh,f^a=fghi,f^b=fi,f^c=gi,g^b=gj,g^c=fgh,i^a=ij,i^c=j,j^c=ij

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213204, #72218292, #72218303
maximal normal subgroup of	C₂×#72213204, #72218292, #72218303
maximal quotient of	#72213209, C₂×#72213204, #72215148, …

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