#72226462

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1439

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂³×C₅²	C₂²×C₄
Fitting	C₂²×C₄×C₅²	C₂×C₄
prime radical	C₂³×D₅²	C₂²
cosocle	C₂³×C₅⋊D₅	C₂³
hypercenter	C₂²×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#1439
LCS	G ▷ C₂³×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

1978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^7,5^2|a^2=d,b^2=df,c^2=df,b^a=bf,c^a=cg,c^b=ce,h^a=h^2,h^b=h^3,h^c=h^3,h^d=h^4,i^a=h^3i^3,i^b=i^3,i^c=i^3,i^d=i^4

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72226463, #72226464, #72226476

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