#72220596

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

#7630939 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	#72220591
commutator, cosocle	C₂³×C₂²⋊A₄	C₂³
Frattini	C₂⁵	C₂²×S₄
Fitting	#7630939	D₃
socle	C₂³	#85546
hypercenter	C₂×D₄	#1020⋊C₂
hypocenter	C₂²⋊A₄	B(4,𝔽₂)
LCS	G ▷ C₂³×C₂²⋊A₄ ▷ C₂×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

11266 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 11266 conjugacy classes of subgroups — too many to draw. Only its 68 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:

2^6,3,2^4|b^2=d,b^a=be,c^a=cef,c^b=cd,d^a=df,e^b=ef,g^b=g^2,g^c=g^2,h^a=hj,h^b=hijk,h^c=hi,h^g=i,i^a=ik,i^b=ik,i^g=hi,j^b=jk,j^c=jk,j^g=k,k^g=jk

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72222995
maximal quotient of	#72267089

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