#72259662

order	64896
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₁₃² ⋊ C₂×C₃×C₄
Hall	#72259289 ⋊ C₃
Hall	#92788 ⋊ C₄×AGL(1,ℤ/8)
Sylow	C₁₃² ⋊ C₃×C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#72214210
commutator	C₄×C₁₃²	C₂³×C₃×C₄
Frattini	C₄	C₂²×#72210302
Fitting	C₂×D₄×C₁₃²	C₂×C₃×C₄
socle	C₂×C₁₃²	C₂×C₃×C₄×D₄
prime radical	#72259514	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂⁴×C₃
hypercenter	C₂×D₄	#72210302
hypocenter	C₁₃²	C₃×C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

4748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^4,3,2^3,13^2|a^2=g,d^2=f,g^2=h,b^a=bgh,c^a=ch,g^b=gh,i^a=i^12,i^d=i^8,i^e=i^3,i^f=i^12,j^a=i^5j,j^d=j^8,j^e=j^3,j^f=j^12

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

holomorph of

#1055

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, 3, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^12);;
gap> SetConjugate(coll, 9, 4, F.9^8);;
gap> SetConjugate(coll, 9, 5, F.9^3);;
gap> SetConjugate(coll, 9, 6, F.9^12);;
gap> SetConjugate(coll, 10, 1, F.9^5*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^8);;
gap> SetConjugate(coll, 10, 5, F.10^3);;
gap> SetConjugate(coll, 10, 6, F.10^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);