#72342826

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72317340 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₈	C₂²×#120⋊C₃
commutator, hypocenter	H(5,𝔽₂)	C₂³×C₃×C₄
Frattini	C₂²×C₄	C₂⁴×A₄
Fitting	#72317340	C₃
socle	C₂	C₂²×C₄×#120⋊C₃
prime radical	#72236824	C₂
cosocle	#585	C₂⁴×C₃
hypercenter	#131	#120⋊C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₈ ◁ #131

14665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342856
maximal normal subgroup of	#72342856

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