#72209883

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#2032030 ⋊ C₃
derived	#15947 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#3532⋊C₃
commutator, cosocle, hypocenter	#15947	C₂×C₃
Frattini, socle	C₂⁴	C₂×C₂²⋊A₄
Fitting	#2032030	C₃
LCS	G ▷ #15947
UCS	C₁ ◁ C₂²

4165 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 4165 conjugacy classes of subgroups — too many to draw. Only its 35 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:

2,3,2^8|c^2=i,d^2=gh,c^a=ch,c^b=cd,d^a=dghi,d^b=ch,d^c=dg,e^b=f,e^c=eij,e^d=egh,f^b=ef,f^c=fi,f^d=fgij,h^b=gi,i^b=hi,j^b=ghij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209885
maximal normal subgroup of	#72209885
maximal quotient of	#72227916, #72228000, #72255743

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