#72214257

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂	#72214227
commutator	C₂×C₄×C₄²⋊C₃	C₂×C₄
Frattini	C₂³×D₄	C₂×S₄
Fitting	#6342426	D₃
socle	C₂³	#73145
prime radical	#72213771	C₂
cosocle	C₂×C₄⋊(#127⋊C₃)	C₂²
hypercenter	C₄²	#4626
hypocenter	C₄²⋊C₃	#353
LCS	G ▷ C₂×C₄×C₄²⋊C₃ ▷ C₂²×C₄²⋊C₃ ▷ C₂×C₄²⋊C₃ ▷ C₄²⋊C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

2834 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 2834 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:

2^6,3,2^4|a^2=c,b^2=e,d^2=f,h^2=j,i^2=k,b^a=bd,c^b=ce,d^a=def,d^c=df,e^a=ef,g^a=g^2,h^a=ik,h^c=hj,h^g=hi,i^a=h,i^c=ik,i^g=hk,j^a=k,j^g=jk,k^a=j,k^g=j

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214317, #72219757, #72227463
maximal normal subgroup of	#72219757
maximal quotient of	#72268885

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