#72259239

order	4992
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×AGL(1,ℤ/8) ⋊ C₃×C₄
Hall	#13130836 ⋊ C₃
Hall	C₁₃⋊C₃ ⋊ #1883
Sylow	C₁₃ ⋊ C₃×#1883

Subgroups

	subgroup	G/H
center	C₂	#72229010
commutator	C₂×C₄×C₁₃	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₁₃)
Fitting	C₁₃×AGL(1,ℤ/8)	C₃×C₄
socle	C₂×C₁₃	C₃×#386
prime radical	AGL(1,ℤ/8)×D₁₃⋊C₃	C₂
cosocle	C₂×C₄×D₁₃	C₂³×C₃
hypercenter	AGL(1,ℤ/8)	AGL(1,𝔽₁₃)
hypocenter	C₁₃	C₃×#1883
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^3,3,2^4,13|c^2=e,g^2=h,b^a=bg,c^a=ch,c^b=cf,f^a=fh,g^a=gh,g^b=gh,g^c=gh,i^c=i^8,i^d=i^3,i^e=i^12

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#453×AGL(1,𝔽₁₃)
maximal normal subgroup of	#453×AGL(1,𝔽₁₃)

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