#72210751

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²	C₂×#67992
commutator, hypocenter	#1273⋊C₃	C₂×C₄
Frattini	C₂²×C₄²	C₂×S₄
Fitting	C₂×#8131	D₃
socle	C₂⁴	C₂×#857
prime radical, cosocle	C₂×#1273⋊C₃	C₂²
hypercenter	C₂×C₄	#67992
LCS	G ▷ #1273⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

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

normal subgroups only

This group has 877 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

Polycyclic presentation:

2^3,3,2^7|a^2=c,b^2=i,c^2=j,e^2=g,f^2=h,g^2=j,h^2=jk,b^a=bi,d^a=d^2ghj,d^c=dghj,e^a=egj,e^d=fgik,f^a=ef,f^d=efi,f^e=fi,g^a=gj,g^d=hj,h^a=ghi,h^d=ghi,j^d=jk,k^a=jk,k^d=j

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210783
maximal normal subgroup of	#72210783

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