#72215409

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

Aliases

No aliases for this group.

Splittings

Fitting

#8813 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11711461
commutator, cosocle	C₂×C₈²⋊C₃	C₂²
Frattini	C₂×C₄²	C₂×S₄
Fitting	#8813	D₃
socle	C₂³	#857⋊C₂
hypocenter	C₈²⋊C₃	D₄
LCS	G ▷ C₂×C₈²⋊C₃ ▷ C₈²⋊C₃
UCS	C₁ ◁ C₂

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

normal subgroups only

This group has 332 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:

2^3,3,2^6|a^2=h,b^2=chj,c^2=j,e^2=g,f^2=h,g^2=i,h^2=j,b^a=bc,c^b=cj,d^a=d^2gi,d^b=dh,d^c=dgi,e^a=efg,e^b=eg,e^d=fgij,f^a=fj,f^b=fh,f^d=ef,g^a=ghi,g^b=gi,g^d=hi,h^b=hj,h^d=gh,i^a=ij,i^d=j,j^d=ij

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218262
maximal normal subgroup of	#72218262
maximal quotient of	#72215414, #72215431, #72215434

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