#72267352

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	C₂×#87588
commutator, cosocle, hypocenter	C₂²⋊#997	C₂³
Frattini	C₂	C₂²×#87588
Fitting	C₂³×#585	D₃
socle	C₂³	C₂²×#1020
hypercenter	C₂×C₄	#87588
LCS	G ▷ C₂²⋊#997
UCS	C₁ ◁ C₄ ◁ C₂×C₄

At least 14500 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has at least 14500 conjugacy classes of subgroups — too many to draw. Only its 98 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:

2^3,3,2^7|b^2=k,c^a=ck,d^c=d^2,e^c=eik,e^d=f,f^c=efjk,f^d=ef,g^d=h,h^c=gh,h^d=gh,i^d=ij,i^f=ik,j^c=ij,j^d=i,j^e=jk,j^f=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267358

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