#72342145

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72342140 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	D₄×#120⋊C₃
commutator, cosocle	#585	C₂⁴×C₃
Frattini	C₂²×C₄	C₂⁴×A₄
Fitting	#72342140	C₃
socle	C₂	C₂×D₄×#120⋊C₃
hypercenter	#135	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₂²×C₃×D₄
LCS	G ▷ #585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

At least 6453 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^6|d^2=fk,f^2=k,g^2=k,h^2=k,c^b=ck,d^b=df,f^b=fk,g^a=gj,g^b=gj,g^e=gh,h^a=hi,h^b=hi,h^e=g,h^g=hk,i^e=j,i^g=ik,j^e=ijk,j^h=jk

Conjugacy classes & character table

124 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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.6*F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 5, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);