#72226467

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #20953

Subgroups

	subgroup	G/H
center	C₂	C₂×#13104114
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₅²×D₄	C₂×C₄
socle	C₂×C₅²	C₂×#386
prime radical	#72221096	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂²×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#20953
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×D₄

5868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Polycyclic presentation:

2^8,5^2|a^2=e,c^2=ef,e^2=h,f^2=h,b^a=bg,c^a=cf,d^a=dh,e^b=eh,e^c=eh,f^b=fh,f^c=fh,g^a=gh,g^c=gh,i^a=i^2,i^c=i^3,i^e=i^4,j^a=i^3j^3,j^c=j^3,j^e=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.5);;
gap> SetPower(coll, 3, F.5*F.6);;
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, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, 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, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 5, 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, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);