#72219936

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×D₄×3₋¹⁺² ⋊ C₂²
Sylow	C₃×3₋¹⁺² ⋊ C₂×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#88668⋊C₃
commutator, cosocle	C₃×C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂²×C₃×D₃²
Fitting	C₂×C₃×D₄×3₋¹⁺²	C₂²
socle	C₂×C₃²	C₂×C₃×D₃×D₄
hypercenter	C₂×D₄	D₃×AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

2484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^2,3^3|a^2=f,f^2=g,h^3=j,b^a=bfg,c^a=cg,f^b=fg,h^a=h^2j^2,h^d=h^2j^2,h^e=hj^2,i^a=ij^2,i^d=i^2,j^a=j^2,j^d=j^2

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219937, #72220090
maximal normal subgroup of	#72220090
maximal quotient of	#72219937

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 3, 3, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 4, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);