#72226505

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅²×D₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×#535

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂⁴×C₄
Fitting	C₂²×C₅²×D₄	C₂×C₄
prime radical	#72221096	C₂
cosocle	C₂²×C₅⋊D₅	C₂⁵
hypercenter	C₂²×D₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#535
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

9732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

172 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.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 3, 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, 3, F.10^3);;
gap> SetConjugate(coll, 10, 6, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);