#72258673

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#62817 ⋊ D₃
derived	#72258498 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₂	#72258287
commutator, cosocle, hypocenter	#72258498	C₂
Frattini, hypercenter	C₂²	#72209700
Fitting	C₂²×#62817	D₃
socle	C₂³	#11715850
LCS	G ▷ #72258498
UCS	C₁ ◁ C₂ ◁ C₂²

At least 11349 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has at least 11349 conjugacy classes of subgroups — too many to draw. Only its 76 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:

2,3,2^10|a^2=l,b^a=b^2,c^a=d,c^b=d,d^a=c,d^b=cd,e^a=f,e^b=f,f^a=e,f^b=ef,g^a=hj,g^b=h,g^d=gk,h^a=gi,h^b=gh,h^c=hk,i^a=j,i^b=j,i^d=il,j^a=i,j^b=ij,j^c=jl,k^a=kl

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258797, #72258952
maximal normal subgroup of	#72258797, #72258952

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