#72243033

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#12705 ⋊ C₄
Sylow	C₅ ⋊ #72242806

Subgroups

	subgroup	G/H
center	C₂	#72242763
commutator	C₂²×C₅×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#12705	C₄
socle	C₂×C₅	#392603
prime radical	D₅×#3484	C₂²
cosocle	C₂²×D₄×D₅	C₂⁴
hypercenter	#12705	AGL(1,𝔽₅)
hypocenter	C₅	#72242806
LCS	G ▷ C₂²×C₅×D₄ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #12705

13270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 80

Polycyclic presentation:

2^10,5|a^2=j,b^2=f,d^2=e,h^2=j,b^a=bg,c^a=cj,c^b=cfghj,d^a=dgi,d^b=dghj,d^c=dfghj,e^a=ej,f^a=fi,g^b=gi,g^c=gj,g^f=gj,h^a=hij,h^b=hi,h^c=hj,h^f=hj,h^g=hj,i^b=ij,i^d=ij,k^d=k^3,k^e=k^4

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#511790
maximal normal subgroup of	AGL(1,𝔽₅)×#511790

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