#72267832

order	5120
automorphism	?
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#72267796 ⋊ C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#17102⋊C₅
commutator, Fitting, cosocle, hypocenter	#72267796	C₅
Frattini, socle	C₂⁶	C₂⁴⋊C₅
LCS	G ▷ #72267796
UCS	C₁ ◁ C₂²

4310 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 4310 conjugacy classes of subgroups — too many to draw. Only its 12 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) = 80

Polycyclic presentation:

5,2^10|b^2=f,c^2=h,d^2=fg,e^2=fhj,b^a=cfh,c^a=ce,c^b=cj,d^a=bdefhk,d^b=dfhjk,d^c=dfhi,e^a=cdeghk,e^b=ef,e^c=efi,e^d=efg,f^a=h,g^a=hk,h^a=ij,i^a=gi,j^a=fi,k^a=f

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267833, #72267872
maximal subgroup of	#72267833, #72267872
maximal normal subgroup of	#72267833, #72267872

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", "k");;
gap> coll := SingleCollector(F, [5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6*F.7);;
gap> SetPower(coll, 5, F.6*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.2*F.4*F.5*F.6*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.3*F.4*F.5*F.7*F.8*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 1, F.8);;
gap> SetConjugate(coll, 7, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.6*F.9);;
gap> SetConjugate(coll, 11, 1, F.6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);