#72226389

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#496 ⋊ C₄
Sylow	C₅² ⋊ #23202

Subgroups

	subgroup	G/H
center	C₂	#72223143
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×#1044
Fitting	C₅²×#496	C₄
socle	C₂×C₅²	#2366
prime radical	C₅⋊D₅×#496	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	#496	#1044
hypocenter	C₅²	#23202
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #496

2588 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:

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

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=g,c^2=f,g^2=h,b^a=bg,c^a=cgh,c^b=ch,d^b=dgh,d^c=df,f^a=fh,g^a=gh,g^d=gh,i^a=i^2,i^e=i^4,j^a=i^3j^3,j^e=j^4

Conjugacy classes & character table

148 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.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, 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, 5, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10^3);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);