#72267334

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72267291 ⋊ C₃
derived	#355894 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#1437⋊C₃
commutator, cosocle, hypocenter	#355894	C₂×C₃
Frattini	C₂⁵×C₄	C₂×A₄
Fitting	#72267291	C₃
socle	C₂²	#7555⋊C₃
LCS	G ▷ #355894
UCS	C₁ ◁ C₂×C₄

6140 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 6140 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 32

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=ijk,g^2=k,h^2=k,j^2=k,c^a=cef,c^b=di,d^a=de,d^b=cdgi,d^c=dgi,e^b=fghk,e^c=eghk,f^b=efghk,f^c=fghk,f^d=fghk,g^b=ij,g^d=gk,h^b=ghijk,h^d=hk,i^b=gjk,i^d=ik,j^b=g,j^c=jk,j^d=jk

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72267348, #72267522

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