#72221188

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×#120 ⋊ C₂
Sylow	C₅² ⋊ #3484

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁵
Fitting	C₂×C₅²×#120	C₂
hypercenter	C₂×#120	C₅⋊D₅
hypocenter	C₅²	#3484
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×#120

7600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#3484
maximal normal subgroup of	C₅⋊D₅×#3484

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