#72259347

order	32448
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Hall	#72259204 ⋊ C₃
Hall	#92788 ⋊ C₄×M₁₆
Sylow	C₁₃² ⋊ C₃×C₄×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂×#72210302
commutator, socle	C₂×C₁₃²	C₂×C₃×C₄²
Fitting	C₂×C₄×C₁₃²	C₂×C₃×C₄
prime radical	C₂²×#11759815	C₂×C₄
cosocle	C₄×C₁₃⋊D₁₃	C₂³×C₃
hypercenter	C₂×C₄	#72210302
hypocenter	C₁₃²	C₃×C₄×M₁₆
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^3,3,2^3,13^2|a^2=f,c^2=e,f^2=g,b^a=bg,h^a=h^12,h^c=h^8,h^d=h^3,h^e=h^12,i^a=h^2i,i^c=i^8,i^d=i^3,i^e=i^12

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259662
maximal normal subgroup of	#72259662

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> SetConjugate(coll, 8, 3, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^8);;
gap> SetConjugate(coll, 9, 4, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);