#72226604

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄²×C₅² ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×#532

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂⁴×C₄
Frattini	C₂²	C₂³×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₄²×C₅²	C₂×C₄
prime radical	#72221334	C₂
cosocle	C₂²×C₅⋊D₅	C₂⁵
hypercenter	C₂×C₄²	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#532
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

8324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 5 generators on 22 points:

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

Polycyclic presentation:

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