Finite Group Databasebeta

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#72237014

order4096
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216698
commutator#35527C22×C4
Frattini, cosocle#6628901C23
prime radical#72236986C2
LCSG ▷ #35527 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6628901 ◁ G
LECSG ▷ #6628901 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1

22346 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 22346 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 32

Polycyclic presentation:

2^12|a^2=jl,b^2=jkl,c^2=d,e^2=l,f^2=k,g^2=l,c^a=cf,c^b=ce,d^a=dg,d^b=dh,e^a=ejl,e^b=ek,e^c=ehl,f^c=fgk,f^d=fi,f^e=fjl,g^b=gj,g^c=gi,g^d=gl,g^e=gl,h^a=hjl,h^b=hl,h^f=hk,h^g=hl,i^b=ik,i^c=il,i^e=il,j^c=jk,j^d=jl,k^c=kl

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72241374, #72338882, #72338996, …
maximal normal subgroup of#72241374, #72338882, #72338996, …
automorphism group of#1829

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10*F.12);;
gap> SetPower(coll, 2, F.10*F.11*F.12);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.12);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);