#72217104

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

Aliases

No aliases for this group.

Splittings

Fitting	#72216292 ⋊ D₃
derived	#11710730 ⋊ C₂³

Subgroups

	subgroup	G/H
center, socle	C₂	#72217073
commutator, cosocle, hypocenter	#11710730	C₂³
Frattini	C₂⁴	C₂²×#1020
Fitting	#72216292	D₃
hypercenter	C₂²	#72215837
LCS	G ▷ #11710730
UCS	C₁ ◁ C₂ ◁ C₂²

At least 9349 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has at least 9349 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^8|e^2=l,f^2=l,d^a=d^2,e^a=el,e^b=ejl,e^c=ejl,e^d=f,f^a=ef,f^b=fil,f^c=fil,f^d=efl,f^e=fl,g^a=gh,g^b=gij,g^d=h,g^e=gk,g^f=gk,h^b=hj,h^d=gh,h^f=hk,i^a=ij,i^d=ij,i^e=il,j^d=i,j^f=jl,k^b=kl

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72218792
maximal quotient of	#72252390

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