#72243091

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#23897 ⋊ C₄
Sylow	C₅ ⋊ #72235423

Subgroups

	subgroup	G/H
center	C₂	#72242760
commutator	C₂²×C₅×D₄	C₂³×C₄
Frattini	C₂²×D₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#23897	C₄
socle	C₂×C₅	#392658
prime radical	D₅×#23897	C₂
cosocle	C₂²×D₄×D₅	C₂⁴
hypercenter	#23897	AGL(1,𝔽₅)
hypocenter	C₅	#72235423
LCS	G ▷ C₂²×C₅×D₄ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #23897

At least 6492 conjugacy classes of subgroups; only the 508 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,5|a^2=ij,b^2=e,d^2=f,h^2=j,b^a=bi,c^a=cj,d^a=dh,d^b=dg,d^c=df,f^a=fj,g^a=gi,h^b=hi,h^c=hj,h^g=hj,i^d=ij,k^b=k^3,k^e=k^4

Conjugacy classes & character table

146 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.9*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 11, 2, F.11^3);;
gap> SetConjugate(coll, 11, 5, F.11^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);