#72221631

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#120 ⋊ C₂²
Sylow	C₅² ⋊ C₂×#525

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁵
Fitting	C₅²×#120	C₂²
hypercenter	#120	D₅²
hypocenter	C₅²	C₂×#525
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #120

4422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221683, #72221698
maximal normal subgroup of	#72221683, #72221698

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