#72226480

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #10954

Subgroups

	subgroup	G/H
center	C₂	#72226473
commutator	C₂×C₄×C₅²	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×#117	C₂×C₄
socle	C₂×C₅²	#1437
prime radical	C₂×#11717471	C₂²
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	#117	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#10954
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #117

2628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 26 points:

(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,18,20,19)(22,23,26,25);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,15,9,16)(18,19,20,21)(23,25,26,24);(1,2,4,8,6,10,5,9)(3,11,12,16,14,7,13,15)(18,19,20,21)(23,24,26,25)

Polycyclic presentation:

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

Conjugacy classes & character table

142 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, 2, F.4*F.5*F.8);;
gap> SetPower(coll, 3, F.4*F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, 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, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
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, 2, 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);