#72343968

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#450 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×S₄
commutator, cosocle	C₂×C₄×A₄	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₂²×#450	D₃
socle	C₂⁴	C₂×D₃×D₄
hypercenter	#450	S₄
hypocenter	A₄	#3003
LCS	G ▷ C₂×C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #450

6681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^7,3,2^2|d^2=fg,f^2=g,d^a=df,d^b=dfg,d^c=de,f^a=fg,f^b=fg,h^b=h^2,h^d=h^2,i^b=ij,i^d=ij,i^h=j,j^h=ij

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72343985

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