#72223334

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1437

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂×D₅⋊AGL(1,𝔽₅)
commutator	C₂²×C₅²	C₂×C₄²
Fitting	C₂⁴×C₅²	C₂×C₄
socle	C₂³×C₅²	C₂²×C₄
prime radical	C₂⁴×C₅⋊D₅	C₂²
cosocle	C₂³×C₅⋊D₅	C₂³
hypercenter	C₂⁴	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#1437
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂⁴

2330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=f,c^2=d,b^a=bg,c^b=ce,h^a=h^4,h^c=h^3,h^d=h^4,i^a=h^2i,i^c=i^3,i^d=i^4

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72223335, #72223345, #72223353, …

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