#72242884

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#23900 ⋊ C₄
Sylow	C₅ ⋊ #72219252

Subgroups

	subgroup	G/H
center	C₂²	#13022440
commutator	C₂³×C₅×D₄	C₂²×C₄
Frattini	C₂³×D₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#23900	C₄
socle	C₂²×C₅	#10991
prime radical	D₅×#23900	C₂
cosocle	C₂³×D₄×D₅	C₂³
hypercenter	#23900	AGL(1,𝔽₅)
hypocenter	C₅	#72219252
LCS	G ▷ C₂³×C₅×D₄ ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #23900

At least 6814 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10,5|b^2=d,c^2=j,b^a=bg,c^a=cf,c^b=ce,e^a=ei,f^b=fh,f^e=fj,g^c=ghij,h^c=hj,i^c=ij,k^b=k^3,k^d=k^4

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#144366
maximal normal subgroup of	AGL(1,𝔽₅)×#144366

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