#72221555

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×D₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₂×#421

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅×C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅×AGL(1,𝔽₅)
Fitting	C₂×C₅²×D₄	C₂×C₄
socle	C₂×C₅²	C₂²×C₂²⋊C₄
prime radical	C₂×#11717477	C₂
cosocle	C₄×C₅×D₅	C₂⁴
hypercenter	C₂×D₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#421
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221622, #72221632, #72221642, …
maximal normal subgroup of	#72221622, #72221632, #72221642, …

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