#72259261

order	2704
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃² ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂	#13062579
commutator, Fitting, cosocle	C₄×C₁₃²	C₂²
Frattini, hypercenter	C₄	D₁₃²
socle	C₂×C₁₃²	D₄
hypocenter	C₁₃²	D₈
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^4,13^2|a^2=c,c^2=d,b^a=bc,c^b=cd,e^a=e^12,e^b=e^12,f^a=e^2f,f^b=f^12

Conjugacy classes & character table

229 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259265, #72259275, #72259276, …
maximal normal subgroup of	#72259265, #72259275, #72259276, …

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