#72226423

order	6400
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₄×#339

Subgroups

	subgroup	G/H
center	C₂×C₄	#11717363
commutator	C₄×C₅²	C₄³
Frattini	C₂²×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₄²×C₅²	C₂×C₄
socle	C₂²×C₅²	C₄×C₂²⋊C₄
prime radical	C₂³×C₅⋊D₅	C₂×D₄
cosocle	C₂²×C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄²	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#339
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

3700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 22 points:

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

Polycyclic presentation:

2^8,5^2|a^2=f,b^2=e,c^2=d,f^2=h,b^a=bfg,e^a=eh,g^a=gh,i^a=i^4,i^c=i^3,i^d=i^4,j^a=i^3j,j^c=j^3,j^d=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.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);