#72227296

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#8926711 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂³	C₂²⋊(C₂²⋊A₄)
commutator, Fitting, cosocle, hypocenter	#8926711	C₃
LCS	G ▷ #8926711
UCS	C₁ ◁ C₂³

2086 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 2086 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

3,2^9|b^2=j,c^2=j,b^a=c,c^a=bcj,c^b=cj,d^a=de,d^b=dh,d^c=dh,e^a=d,e^b=eh,f^a=g,f^b=fhi,f^d=fj,g^a=fg,g^b=ghi,g^c=ghi,g^e=gj

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227713, #72227717, #72227728
maximal subgroup of	#72227299, #72227713, #72227717, …
maximal normal subgroup of	#72227299, #72227713, #72227717, …
maximal quotient of	#72267935, #72268095, #72268380

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