#72226552

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #10765

Subgroups

	subgroup	G/H
center	C₂	#72226477
commutator	C₂×C₄×C₅²	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×#41	C₂×C₄
socle	C₂×C₅²	C₂²⋊#95
prime radical	C₂×#41×C₅⋊D₅	C₂²
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	C₂×#41	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#10765
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#41

4148 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:

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

Polycyclic presentation:

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