#72223213

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₅

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#1357

Subgroups

	subgroup	G/H
center	C₂	#72223212
commutator	C₄²×C₅²	C₂²×C₄
Frattini	C₄²	C₂²×#1044
Fitting	C₅²×#493	C₄
socle	C₂×C₅²	C₂×#331
prime radical	C₅⋊D₅×#493	C₂
cosocle	C₄×#6029	C₂³
hypercenter	#493	#1044
hypocenter	C₅²	C₂×#1357
LCS	G ▷ C₄²×C₅² ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

1748 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 122 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,c^2=gh,d^2=fh,e^2=fg,f^2=h,g^2=h,b^a=be,c^a=cg,c^b=cg,d^b=df,e^a=eg,e^b=efg,f^b=fh,g^a=gh,g^b=gh,i^a=i^2,i^d=i^4,j^a=i^3j^3,j^d=j^4

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

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