#72218214

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting

C₄×#1842 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	#11713252
commutator	C₄×C₄²⋊C₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×S₄
Fitting	C₄×#1842	D₃
socle	C₂³	C₄×C₄⋊S₄
prime radical	#72213885	C₂
cosocle	C₂×C₄×C₄²⋊C₃	C₂³
hypercenter	C₄²	#857⋊C₂
hypocenter	C₄²⋊C₃	#443
LCS	G ▷ C₄×C₄²⋊C₃ ▷ C₂×C₄²⋊C₃ ▷ C₄²⋊C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

1702 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 1702 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|a^2=k,b^2=efk,c^2=d,e^2=f,h^2=j,i^2=k,b^a=be,c^a=cek,d^a=df,e^a=ef,g^a=g^2j,g^b=gjk,g^e=gj,h^a=hij,h^b=hj,h^g=ij,i^b=ik,i^g=hi,j^a=jk,j^g=k,k^g=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218458
maximal quotient of	#72218219

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