#72223161

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄²×C₅² ⋊ C₂²
Sylow	C₅² ⋊ #3018

Subgroups

	subgroup	G/H
center	C₄	#11717041
commutator, cosocle	C₂×C₄×C₅²	C₂⁴
Frattini	C₂×C₄	C₂²×D₅²
Fitting	C₂×C₄²×C₅²	C₂²
socle	C₂×C₅²	C₂×#120
hypercenter	C₂×C₄²	D₅²
hypocenter	C₅²	#3018
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223172, #72226483, #72227631
maximal normal subgroup of	#72223172, #72226483, #72227631

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, 2, F.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*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.8^4);;
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);