#72223289

order	12800
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#1715 ⋊ C₄
Sylow	C₅² ⋊ #213219

Subgroups

	subgroup	G/H
center	C₂²	#127×#1043
commutator	C₄²×C₅²	C₂³×C₄
Frattini	C₄²	C₂³×#1043
Fitting	C₅²×#1715	C₄
socle	C₂²×C₅²	C₄×#127
prime radical	C₅⋊D₅×#493	C₂²
cosocle	C₄²×C₅⋊D₅	C₂⁴
hypercenter	#1715	#1043
hypocenter	C₅²	#213219
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1715

7888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^9,5^2|a^2=e,c^2=efh,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^b=dg,d^c=di,e^b=eh,f^b=fh,g^b=gi,j^a=j^3,j^c=j^3,j^e=j^4,k^a=k^3,k^c=k^3,k^e=k^4

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

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