#72221078

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₄	C₂×#87830
commutator	C₄×C₅²	C₂×C₄²
Frattini	C₂×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅²	C₂×C₄
socle	C₂×C₅²	C₂×#95
prime radical	C₂³×C₅⋊D₅	D₄
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	C₂²×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=e,b^2=d,e^2=g,b^a=bef,c^a=cg,d^a=dg,e^b=eg,f^b=fg,h^a=h^4,h^b=h^3,h^d=h^4,i^a=h^3i,i^b=i^3,i^d=i^4

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223340, #72223353, #72223365, …
maximal normal subgroup of	#72223340, #72223353, #72223365, …

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