#72226455

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #10997

Subgroups

	subgroup	G/H
center	C₂²	#13104117
commutator	C₂²×C₄×C₅²	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₅²×D₄	C₂×C₄
socle	C₂²×C₅²	#379
prime radical	#72221096	C₂
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	C₂²×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#10997
LCS	G ▷ C₂²×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

4620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 22 points:

(6,11)(7,8)(10,12)(13,14,16,15)(18,19,22,21);(2,4)(6,10)(7,8);(1,2)(3,4)(5,6,7,11,9,10,8,12)(14,15,16,17)(19,20,22,21)

Polycyclic presentation:

2^8,5^2|a^2=d,c^2=df,d^2=h,f^2=h,b^a=bg,c^a=cf,c^b=ce,d^b=dh,d^c=dh,e^a=eh,f^b=fh,f^c=fh,g^a=gh,i^a=i^2,i^c=i^3,i^d=i^4,j^a=i^3j^3,j^c=j^3,j^d=j^4

Conjugacy classes & character table

136 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.4*F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8);;
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, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10^3);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);