#72275641

order	8000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×5₋¹⁺² ⋊ C₄
Sylow	5₋¹⁺² ⋊ #390

Subgroups

	subgroup	G/H
center	C₂²	C₂²×AGL(1,ℤ/25)
commutator	C₂²×C₂₅	C₂²×C₄×C₅
Frattini, socle	C₂²×C₅	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₂×D₄×5₋¹⁺²	C₄
prime radical	C₂×D₄×D₂₅⋊C₅	C₂
cosocle	C₂²×D₂₅	C₂³×C₅
hypercenter	C₂×D₄	AGL(1,ℤ/25)
hypocenter	C₂₅	C₅×#390
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

801 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 400

Polycyclic presentation:

2^3,5,2^3,5^2|b^2=e,c^2=g,h^5=i,b^a=bf,c^a=cg,c^b=cg,h^b=h^3i^3,h^d=hi^3,h^e=h^4i^4,i^b=i^3,i^e=i^4

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#127×AGL(1,ℤ/25)
maximal normal subgroup of	#127×AGL(1,ℤ/25)

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