#72226422

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #10895

Subgroups

	subgroup	G/H
center	C₄	#13104001
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	#10895
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

2284 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)(4,12)(5,11)(6,9)(8,15)(10,13)(14,16)(17,18,20,19)(22,23,24,25);(1,3)(2,6,8,14)(4,12)(5,11)(7,9,15,16)(10,13)(18,19,20,21)(23,25,24,26);(1,2,4,9,5,8,13,16)(3,6,10,7,11,14,12,15)(23,24)(25,26)

Polycyclic presentation:

2^8,5^2|a^2=f,b^2=d,c^2=de,f^2=h,b^a=bg,c^a=cfh,d^a=dh,e^a=eh,f^b=fh,f^c=fh,g^a=gh,g^b=gh,g^c=gh,i^a=i^4,i^b=i^3,i^c=i^3,i^d=i^4,j^a=i^3j,j^b=j^3,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.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, 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, 7, 2, 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, 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);;
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);