#72224446

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72210047 ⋊ C₃
derived	C₂²×#2679 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209713
commutator, cosocle, hypocenter	C₂²×#2679	C₂×C₃
Frattini, socle	C₂⁵	C₂×C₂²⋊A₄
Fitting	#72210047	C₃
LCS	G ▷ C₂²×#2679
UCS	C₁ ◁ C₂

46135 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 46135 conjugacy classes of subgroups — too many to draw. Only its 38 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:

2,3,2^9|e^2=ik,f^2=j,c^b=cd,d^b=c,e^a=egh,e^b=fjk,e^c=ei,e^d=ek,f^a=fg,f^b=efij,f^c=fik,f^d=fj,f^e=fij,g^b=h,h^b=gh,i^b=ijk,j^b=k,k^b=ik

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234286
maximal normal subgroup of	#72234286
maximal quotient of	#72224447, #72255743, #72267233, …

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