#72340745

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72340744 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	#72340508
commutator, cosocle	#52301	C₂³×C₃
Frattini	C₂²×C₄	C₂³×C₂²⋊A₄
Fitting	#72340744	C₃
socle	C₂³	C₂×D₄×C₂²⋊A₄
hypercenter	C₈	#63011⋊C₃
hypocenter	#2679	C₂×C₃×D₄
LCS	G ▷ #52301 ▷ #2679
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

35279 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 35279 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^3,3,2^8|c^2=ekl,e^2=kl,f^2=k,g^2=j,c^b=ce,d^b=djkl,e^b=ekl,f^a=fl,f^b=fjkl,f^d=fg,g^a=gjkl,g^b=gjk,g^d=fjkl,g^f=gkl,h^a=hjkl,h^d=i,h^f=hjkl,h^g=hk,i^a=il,i^d=hi,i^f=ijl,i^g=ij,j^d=k,k^d=jl,l^d=jk

Conjugacy classes & character table

118 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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5*F.11*F.12);;
gap> SetPower(coll, 5, F.11*F.12);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.6*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.11*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10*F.12);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.11);;
gap> SetConjugate(coll, 11, 4, F.10*F.12);;
gap> SetConjugate(coll, 12, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);