#72269864

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#23996 ⋊ C₄
Sylow	C₅ ⋊ #72245896

Subgroups

	subgroup	G/H
center	C₂	#72269428
commutator	C₂⁴×C₅	C₂⁴×C₄
Frattini	C₂⁴	C₂⁴×AGL(1,𝔽₅)
Fitting	C₅×#23996	C₄
socle	C₂×C₅	#515668
prime radical	D₅×#23996	C₂
cosocle	C₂⁴×D₅	C₂⁵
hypercenter	#23996	AGL(1,𝔽₅)
hypocenter	C₅	#72245896
LCS	G ▷ C₂⁴×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #23996

At least 8836 conjugacy classes of subgroups; only the 1202 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,5|a^2=j,b^2=f,c^2=i,b^a=bg,c^a=cgj,c^b=ci,e^a=eh,e^d=ei,g^c=gj,h^d=hj,i^a=ij,k^b=k^3,k^f=k^4

Conjugacy classes & character table

176 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, 2, 5]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 11, 2, F.11^3);;
gap> SetConjugate(coll, 11, 6, F.11^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);