#72223285

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #3585

Subgroups

	subgroup	G/H
center	C₂	C₂³×#6038
commutator, cosocle	C₄×C₅²	C₂⁵
Frattini	C₄	C₂³×D₅²
Fitting	C₂×C₅²×#41	C₂²
socle	C₂×C₅²	C₂³×D₄
hypercenter	C₂×#41	D₅²
hypocenter	C₅²	#3585
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×#41

3826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

257 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223301
maximal normal subgroup of	#72223301

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