#72220191

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×D₈ ⋊ SD₁₆
Sylow	C₃² ⋊ #19803

Subgroups

	subgroup	G/H
center	C₂	D₄×AΓL(1,𝔽₉)
commutator, cosocle	C₄×#157	C₂⁴
Frattini	C₄	C₂²×AΓL(1,𝔽₉)
Fitting	C₃²×D₈	SD₁₆
socle	C₂×C₃²	D₄×SD₁₆
prime radical	#11885073	C₂
hypercenter	D₈	AΓL(1,𝔽₉)
hypocenter	C₃²	#19803
LCS	G ▷ C₄×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=e,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cg,d^c=df,e^b=eg,f^c=fh,f^d=fh,i^c=i^2,i^d=ij,i^f=i^2j,i^h=i^2,j^c=ij,j^d=ij^2,j^f=ij,j^h=j^2

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220192, AGL(1,ℤ/8)×AΓL(1,𝔽₉)
maximal normal subgroup of	AGL(1,ℤ/8)×AΓL(1,𝔽₉)

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