#72253131

order	12800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#1368 ⋊ C₄
Sylow	C₅² ⋊ #129165

Subgroups

	subgroup	G/H
center	C₂²	#72220668
commutator	C₅²×C₈	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×#1043
Fitting	C₅²×#1368	C₄
socle	C₂²×C₅²	#1776
prime radical	C₂²×C₅⋊D₅	#442
cosocle	C₄×C₈×C₅⋊D₅	C₂³
hypercenter	#1368	#1043
hypocenter	C₅²	#129165
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 3200

Polycyclic presentation:

2^9,5^2|a^2=d,b^2=de,c^2=f,d^2=g,f^2=h,h^2=i,c^a=cf,c^b=ch,d^c=di,f^a=fh,f^b=fi,h^a=hi,j^b=j^3,j^e=j^4,k^b=k^3,k^e=k^4

Conjugacy classes & character table

284 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1043×#12067
maximal normal subgroup of	#1043×#12067

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