#72258607

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×AGL(1,ℤ/8) ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×S₄
commutator, cosocle	C₄×A₄	C₂⁵
Frattini	C₄	C₂⁴×S₄
Fitting	C₂³×AGL(1,ℤ/8)	D₃
socle	C₂³	C₂²×D₃×D₄
hypercenter	C₂×AGL(1,ℤ/8)	S₄
hypocenter	A₄	#3585
LCS	G ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

10535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^7,3,2^2|a^2=f,f^2=g,b^a=bfg,c^a=cg,e^d=eg,f^b=fg,h^a=h^2,h^d=h^2,i^a=j,i^d=j,i^h=ij,j^a=i,j^d=i,j^h=i

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72258631, AGL(1,ℤ/8)×GL(2,ℤ/4)

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