#72226495

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#532 ⋊ C₄
Sylow	C₅² ⋊ C₂×#1884

Subgroups

	subgroup	G/H
center	C₄	#13104917
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×#1044
Fitting	C₅²×#532	C₄
socle	C₂×C₅²	C₂×#386
prime radical	C₅⋊D₅×#532	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	#532	#1044
hypocenter	C₅²	C₂×#1884
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #532

5316 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:

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

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=f,e^2=h,g^2=h,b^a=bg,c^b=cf,d^a=dh,d^c=dh,f^a=fh,g^a=gh,g^c=gh,i^a=i^2,i^e=i^4,j^a=i^3j^3,j^e=j^4

Conjugacy classes & character table

232 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.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 6, 1, 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, 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);