#72259272

order	5408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₁₃² ⋊ #102

Subgroups

	subgroup	G/H
center	C₂	#72214177
commutator, Fitting	C₄×C₁₃²	C₂×C₄
Frattini, hypercenter	C₄	D₁₃⋊#261
socle	C₂×C₁₃²	C₂²⋊C₄
prime radical	#72229264	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂²
hypocenter	C₁₃²	#102
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^5,13^2|a^2=c,b^2=cde,c^2=e,d^2=e,b^a=bd,c^b=ce,d^b=de,f^a=f^5,f^b=f^8,f^c=f^12,g^a=f^6g^8,g^b=g^8,g^c=g^12

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259273, #72259283, #72259288, …
maximal normal subgroup of	#72259273, #72259283, #72259288, …

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