#72209644

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#15258 ⋊ D₃
derived	C₂²⋊(C₄²⋊C₃) ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²⋊(C₄²⋊C₃)	C₂³
Frattini, socle	C₂⁴	C₂²×S₄
Fitting	#15258	D₃
LCS	G ▷ C₂²⋊(C₄²⋊C₃)
UCS	C₁

2682 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 2682 conjugacy classes of subgroups — too many to draw. Only its 33 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|e^2=j,f^2=i,c^a=ci,d^a=dij,d^b=dj,d^c=d^2,e^a=ej,e^b=eh,e^c=efi,e^d=fj,f^a=fi,f^b=fg,f^c=fi,f^d=ef,g^d=gh,h^c=gh,h^d=g,i^d=ij,j^c=ij,j^d=i

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209646, #72235038, #72237390
maximal quotient of	#72209784, #72209792

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