#72259289

order	21632
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₁₃² ⋊ C₂×C₄
Sylow	C₁₃² ⋊ C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#72214198
commutator	C₄×C₁₃²	C₂³×C₄
Frattini	C₄	C₂²×D₁₃⋊#261
Fitting	C₂×D₄×C₁₃²	C₂×C₄
socle	C₂×C₁₃²	C₂×C₄×D₄
prime radical	#72259277	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂⁴
hypercenter	C₂×D₄	D₁₃⋊#261
hypocenter	C₁₃²	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

2374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^7,13^2|a^2=f,d^2=e,f^2=g,b^a=bfg,c^a=cg,f^b=fg,h^a=h^12,h^d=h^8,h^e=h^12,i^a=h^2i,i^d=i^8,i^e=i^12

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259662
maximal normal subgroup of	#72259662

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