#72221199

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×H(5,𝔽₂) ⋊ C₂
Sylow	C₅² ⋊ #3498

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁵
Fitting	C₂×C₅²×H(5,𝔽₂)	C₂
hypercenter	C₂×H(5,𝔽₂)	C₅⋊D₅
hypocenter	C₅²	#3498
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

6992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|d^2=g,b^a=bf,c^a=cf,d^a=dg,d^b=dg,e^c=eg,h^b=h^4,h^c=h^4,i^b=i^4,i^c=i^4

Conjugacy classes & character table

452 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#3498
maximal normal subgroup of	C₅⋊D₅×#3498

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