#72275822

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

Aliases

No aliases for this group.

Splittings

Fitting	H(5,𝔽₂)×5₋¹⁺² ⋊ C₂
Sylow	5₋¹⁺² ⋊ #453

Subgroups

	subgroup	G/H
center	C₂	#72275637
commutator, cosocle	C₂×C₄×C₂₅	C₂³×C₅
Frattini	C₂×C₄×C₅	C₂²×C₅×D₅
Fitting	H(5,𝔽₂)×5₋¹⁺²	C₂
socle	C₂×C₅	C₅×#4099
hypercenter	H(5,𝔽₂)	D₂₅⋊C₅
hypocenter	C₂₅	C₅×#453
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

670 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 200

Polycyclic presentation:

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

Conjugacy classes & character table

284 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 6, 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.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^4*F.9^4);;
gap> SetConjugate(coll, 8, 4, F.8*F.9^3);;
gap> SetConjugate(coll, 9, 2, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);