#72221731

order	12800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#127 ⋊ C₄²
Sylow	C₅² ⋊ C₄×#1884

Subgroups

	subgroup	G/H
center	C₂	#72221729
commutator	C₂×C₄×C₅²	C₂²×C₄²
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)²
Fitting	C₅²×#127	C₄²
socle	C₂×C₅²	C₄×#386
prime radical	#72221547	C₂²
cosocle	C₂×C₄×D₅²	C₂⁴
hypercenter	#127	AGL(1,𝔽₅)²
hypocenter	C₅²	C₄×#1884
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #127

8382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,5^2|a^2=e,b^2=g,d^2=f,e^2=i,h^2=i,b^a=bh,c^b=cg,g^a=gi,h^a=hi,h^c=hi,j^a=j^2,j^d=j^3,j^e=j^4,j^f=j^4,k^a=jk,k^d=k^3,k^e=j^3k,k^f=k^4

Conjugacy classes & character table

206 conjugacy classes — view the full character table.

Relationships

holomorph of

#700

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 4, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> SetConjugate(coll, 10, 6, F.10^4);;
gap> SetConjugate(coll, 11, 1, F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.11^3);;
gap> SetConjugate(coll, 11, 5, F.10^3*F.11);;
gap> SetConjugate(coll, 11, 6, F.11^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);