#72226573

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₄×#430

Subgroups

	subgroup	G/H
center	C₄	C₂×#11717375
commutator	C₄×C₅²	C₂²×C₄²
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×M₁₆	C₂×C₄
socle	C₂×C₅²	C₂×C₄×C₄⋊C₄
prime radical	C₂³×C₅⋊D₅	C₂×Q₈
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×M₁₆	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#430
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

2724 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:

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

Polycyclic presentation:

2^8,5^2|a^2=g,b^2=g,d^2=e,f^2=h,g^2=h,b^a=bf,c^a=ch,f^a=fh,f^b=fh,i^a=i^4,i^d=i^3,i^e=i^4,j^a=i^3j,j^d=j^3,j^e=j^4

Conjugacy classes & character table

172 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.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
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, 4, F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);