#72222569

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#153709 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11709197
commutator, Fitting, cosocle, hypocenter	#153709	C₃
Frattini, socle	C₂⁵	C₂²⋊A₄
LCS	G ▷ #153709
UCS	C₁ ◁ C₂

1715 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1715 conjugacy classes of subgroups — too many to draw. Only its 41 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:

3,2^9|b^2=f,c^2=g,d^2=h,e^2=i,b^a=bc,c^a=bgj,c^b=cj,d^a=eh,d^b=dj,d^c=dj,e^a=de,e^b=ej,f^a=fgj,g^a=f,h^a=i,i^a=hi

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72228154, #72238916, C₂×#72238916, …
maximal subgroup of	#72222575, #72222672, #72222708, …
maximal normal subgroup of	#72222575, #72222672, #72222708, …
maximal quotient of	C₂×#72222569

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