#72226408

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #11129

Subgroups

	subgroup	G/H
center	C₂	#72223351
commutator	C₂²×C₄×C₅²	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₅²×D₄	C₂×C₄
socle	C₂×C₅²	#1439
prime radical	C₂²×D₄×C₅⋊D₅	C₂²
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	C₂²×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#11129
LCS	G ▷ C₂²×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×D₄

3148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

130 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, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*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, 6, 2, F.6*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.9^4);;
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);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);