#72226530

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #20890

Subgroups

	subgroup	G/H
center	C₂	C₂×#13104122
commutator	C₂³×C₅²	C₂³×C₄
Frattini	C₂³	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×#41	C₂×C₄
socle	C₂×C₅²	C₂×#385
prime radical	C₂×#41×C₅⋊D₅	C₂²
cosocle	C₂³×C₅⋊D₅	C₂⁴
hypercenter	C₂×#41	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#20890
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#41

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:

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

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=eh,d^2=h,b^a=bf,c^b=cfg,d^a=dh,d^b=dh,f^c=fh,g^a=gh,g^c=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.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, 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);