#72267571

order	6144
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#72267566 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	C₂²⋊#5464
commutator, Fitting, cosocle, hypocenter	#72267566	C₃
Frattini, socle	C₂⁵	C₂²⋊(C₂²⋊A₄)
LCS	G ▷ #72267566
UCS	C₁ ◁ C₂³

45298 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 45298 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 384

Polycyclic presentation:

3,2^11|d^2=j,e^2=hijl,f^2=l,g^2=l,b^a=bc,c^a=b,d^a=eh,d^b=dh,d^c=di,e^a=dehj,e^b=ehi,e^c=ehl,e^d=ehj,f^a=fg,f^d=fhjk,f^e=fhjkl,g^a=f,g^d=ghjkl,g^e=gl,g^f=gl,h^a=il,i^a=hi,j^a=hijl

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267589
maximal normal subgroup of	#72267589

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