#72244717

order	6591
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	13₋¹⁺² ⋊ C₃
derived	C₁₆₉ ⋊ C₃×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₁₆₉	C₃×C₁₃
Frattini, socle	C₁₃	C₁₃×C₁₃⋊C₃
Fitting	13₋¹⁺²	C₃
LCS	G ▷ C₁₆₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 169

Polycyclic presentation:

3,13^3|c^13=d,c^a=c^3d^11,c^b=cd^12,d^a=d^3

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244718
maximal normal subgroup of	#72244718

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d");;
gap> coll := SingleCollector(F, [3, 13, 13, 13]);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 3, 1, F.3^3*F.4^11);;
gap> SetConjugate(coll, 3, 2, F.3*F.4^12);;
gap> SetConjugate(coll, 4, 1, F.4^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);