#72247728

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#8108057 ⋊ C₃
derived	#2679 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11710731
commutator, cosocle, hypocenter	#2679	C₂²×C₃
Frattini, socle	C₂³	C₂²×C₂²⋊A₄
Fitting	#8108057	C₃
LCS	G ▷ #2679
UCS	C₁ ◁ C₂

4783 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 4783 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 24

Polycyclic presentation:

2^2,3,2^7|d^2=h,e^2=hj,d^a=dh,d^b=dj,d^c=de,e^a=ehj,e^b=eh,e^c=dhj,f^c=g,f^d=fhi,f^e=fhj,g^c=fg,g^d=gh,g^e=ghi,h^c=j,i^c=ij,j^c=hj

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255786, #72256013, #72340973, …
maximal normal subgroup of	#72256013, #72340973, #72341177

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