#72259253

order	5408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²×C₁₃

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₄
Sylow	C₁₃² ⋊ #102

Subgroups

	subgroup	G/H
center	C₂	#72214162
commutator	C₄×C₁₃²	C₂×C₄
Frattini	C₄	C₂×#10621729
Fitting	D₄×C₁₃²	C₄
socle	C₂×C₁₃²	C₂²⋊C₄
prime radical	D₄×C₁₃⋊D₁₃	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂²
hypercenter	D₄	#10621729
hypocenter	C₁₃²	#102
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259257, #72259273, #72259280, …
maximal normal subgroup of	#72259257, #72259273, #72259280, …

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, 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, 3, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);