#72259573

order	32448
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₂×C₃×C₄
Hall	#72259270 ⋊ C₃
Hall	#92788 ⋊ C₄×#41
Sylow	C₁₃² ⋊ C₃×C₄×#41

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×#72210302
commutator, socle	C₂×C₁₃²	C₂³×C₃×C₄
Fitting	D₄×C₁₃²	C₂×C₃×C₄
prime radical	#72259508	C₂
cosocle	C₂×C₁₃⋊D₁₃	C₂⁴×C₃
hypercenter	D₄	#72210302
hypocenter	C₁₃²	C₃×C₄×#41
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ D₄

1804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^4,3,2^2,13^2|a^2=g,d^2=f,b^a=bg,c^b=cg,h^a=h^12,h^d=h^8,h^e=h^9,h^f=h^12,i^a=h^2i,i^d=i^8,i^e=i^9,i^f=i^12

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

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