#72221646

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#344

Subgroups

	subgroup	G/H
center	C₂	D₅×(C₂×AGL(1,𝔽₅))⋊C₄
commutator	C₂²×C₅²	C₂×C₄²
Frattini	C₂³	C₂×D₅×AGL(1,𝔽₅)
Fitting	C₅²×C₂²⋊C₄	C₂×C₄
socle	C₂×C₅²	C₂×#95
prime radical	C₂³×C₅⋊D₅	C₂×C₄
cosocle	C₂²×C₅×Dic₅	C₂³
hypercenter	C₂²⋊C₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#344
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=d,b^2=e,d^2=g,b^a=bf,e^a=eg,f^b=fg,h^a=h^2,h^c=h^4,h^d=h^4,i^a=h^2i,i^c=i^4,i^d=hi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221698, #72221709, #72221712, …
maximal normal subgroup of	#72221698, #72221709, #72221712, …

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