#72212876

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂	#72209712
commutator, cosocle	#11710913	C₂²
Frattini	C₄	C₂×#87588
Fitting	C₂⁴×2₋¹⁺⁴	D₃
socle	C₂⁵	GL(2,ℤ/4)
prime radical	#72210817	C₂
hypercenter	D₄	#87588
hypocenter	C₂²⋊(C₂²⋊SL(2,𝔽₃))	D₄
LCS	G ▷ #11710913 ▷ C₂²⋊(C₂²⋊SL(2,𝔽₃))
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

75959 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 75959 conjugacy classes of subgroups — too many to draw. Only its 59 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|b^2=k,c^2=k,i^2=k,j^2=k,b^a=bc,c^a=ck,c^b=ck,d^b=d^2,e^b=efh,e^d=f,f^b=fgh,f^d=ef,g^b=h,g^d=gh,h^b=g,h^d=g,i^b=ehjk,i^d=j,j^b=efghik,j^d=ijk,j^i=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213263, C₂×#72212876
maximal normal subgroup of	C₂×#72212876
maximal quotient of	C₂×#72212876, #72267529

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