#72259226

order	2704
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃² ⋊ #41

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₁₃²
commutator, socle, cosocle	C₂×C₁₃²	C₂³
Frattini	C₂	C₂×D₁₃²
Fitting	C₄×C₁₃²	C₂²
hypocenter	C₁₃²	#41
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259229, #72259284, #72259375
maximal normal subgroup of	#72259229, #72259284, #72259375

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