#72226579

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅²×M₁₆ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×#413

Subgroups

	subgroup	G/H
center	C₄	#13104045
commutator	C₂²×C₅²	C₂²×C₄²
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×M₁₆	C₂×C₄
socle	C₂×C₅²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#11717471	C₂²
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×M₁₆	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#413
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

3796 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:

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

Polycyclic presentation:

2^8,5^2|a^2=g,b^2=fg,d^2=e,g^2=h,b^a=bf,c^a=ch,f^a=fh,f^b=fh,g^b=gh,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.6*F.7);;
gap> SetPower(coll, 4, F.5);;
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, 7, 2, F.7*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);