#72226577

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #21076

Subgroups

	subgroup	G/H
center	C₂	C₂×#13103990
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×M₁₆	C₂×C₄
socle	C₂×C₅²	C₂×#388
prime radical	C₂×#11717471	C₂²
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×M₁₆	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#21076
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×M₁₆

2580 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)(6,14)(9,15)(11,12)(17,18,20,19)(22,23,24,25);(2,7)(6,9)(8,16)(14,15);(1,3)(2,6)(4,12)(5,11)(7,9)(8,15)(10,13)(14,16)(18,19,20,21)(23,25,24,26);(1,2,4,8,5,7,13,16)(3,9,12,14,11,6,10,15)(23,24)(25,26)

Polycyclic presentation:

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