#72226581

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #21249

Subgroups

	subgroup	G/H
center	C₂	C₂×#13104115
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₅²×D₄	C₂×C₄
socle	C₂×C₅²	C₂×#390
prime radical	C₂×#13106298	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂²×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#21249
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×D₄

4132 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:

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

Polycyclic presentation:

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