#72223063

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#498 ⋊ C₄
Sylow	C₅² ⋊ #9841

Subgroups

	subgroup	G/H
center	C₂²	#13104792
commutator	C₄²×C₅²	C₂²×C₄
Frattini	C₄²	C₂²×#1043
Fitting	C₅²×#498	C₄
socle	C₂²×C₅²	#390
prime radical	C₂²×C₅⋊D₅	#127
cosocle	C₄²×C₅⋊D₅	C₂³
hypercenter	#498	#1043
hypocenter	C₅²	#9841
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

2040 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 2040 conjugacy classes of subgroups — too many to draw. Only its 282 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:

2^8,5^2|a^2=d,b^2=de,c^2=fh,d^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,c^b=cf,d^b=dg,e^b=eg,f^b=fh,i^a=i^3,i^b=i^3,i^d=i^4,j^a=j^3,j^b=j^3,j^d=j^4

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223260, #72223297, #1043×#1715
maximal normal subgroup of	#72223260, #72223297, #1043×#1715

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