#72226432

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #10893

Subgroups

	subgroup	G/H
center	C₄	#13104048
commutator	C₂×C₄×C₅²	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₄²×C₅²	C₂×C₄
socle	C₂×C₅²	#1436
prime radical	C₂³×C₅⋊D₅	C₂×D₄
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄²	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#10893
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

2180 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)(4,10)(5,11)(6,13)(8,15)(9,14)(12,16)(18,19,20,21)(22,23,24,25);(2,6,8,9)(7,14,15,13)(17,18,21,19,20);(1,2,4,9,5,8,12,6)(3,7,10,14,11,15,16,13)(23,24)(25,26)

Polycyclic presentation:

2^8,5^2|a^2=f,b^2=e,c^2=d,f^2=h,b^a=bfg,c^b=ce,e^a=eh,g^a=gh,g^c=gh,i^a=i^4,i^c=i^3,i^d=i^4,j^a=i^3j,j^c=j^3,j^d=j^4

Conjugacy classes & character table

184 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.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*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.9^4);;
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);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);