Finite Group Databasebeta

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#72340935

order12288
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#72241389 ⋊ D3
derived#72340916 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72340916C2
FrattiniC25#87588
Fitting#72241389D3
socleC22#72340869
LCSG ▷ #72340916
UCSC1

8978 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 8978 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 48

Polycyclic presentation:

2,3,2^11|a^2=klm,c^2=j,d^2=jm,e^2=ijlm,f^2=ijm,g^2=l,h^2=lm,b^a=b^2,c^a=cdm,c^b=dl,d^a=dj,d^b=cdjl,d^c=djl,e^a=cdefil,e^b=flm,e^c=ejk,e^d=ei,f^a=df,f^b=efij,f^c=fijk,f^d=fjklm,f^e=fij,g^a=ghlm,g^b=hl,g^c=gl,g^d=gjlm,g^e=gjlm,g^f=gjk,h^a=hlm,h^b=gh,h^c=hjlm,h^d=hlm,h^e=hkm,h^f=hj,i^a=ik,i^b=im,i^c=im,i^d=il,i^e=il,i^f=ilm,i^g=il,i^h=ilm,j^a=jlm,j^b=jm,j^e=jl,j^f=jlm,k^b=kl,k^c=kl,k^d=klm,k^e=klm,k^f=km,l^a=m,l^b=lm,m^a=l,m^b=l

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

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", "m");;
gap> coll := SingleCollector(F, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11*F.12*F.13);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10*F.13);;
gap> SetPower(coll, 5, F.9*F.10*F.12*F.13);;
gap> SetPower(coll, 6, F.9*F.10*F.13);;
gap> SetPower(coll, 7, F.12);;
gap> SetPower(coll, 8, F.12*F.13);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.13);;
gap> SetConjugate(coll, 3, 2, F.4*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.10*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.12);;
gap> SetConjugate(coll, 5, 1, F.3*F.4*F.5*F.6*F.9*F.12);;
gap> SetConjugate(coll, 5, 2, F.6*F.12*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.4*F.6);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.12*F.13);;
gap> SetConjugate(coll, 7, 2, F.8*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.12*F.13);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.12*F.13);;
gap> SetConjugate(coll, 8, 4, F.8*F.12*F.13);;
gap> SetConjugate(coll, 8, 5, F.8*F.11*F.13);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.13);;
gap> SetConjugate(coll, 9, 3, F.9*F.13);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.12*F.13);;
gap> SetConjugate(coll, 9, 7, F.9*F.12);;
gap> SetConjugate(coll, 9, 8, F.9*F.12*F.13);;
gap> SetConjugate(coll, 10, 1, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.13);;
gap> SetConjugate(coll, 10, 5, F.10*F.12);;
gap> SetConjugate(coll, 10, 6, F.10*F.12*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> SetConjugate(coll, 11, 4, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 5, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 6, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 13, 1, F.12);;
gap> SetConjugate(coll, 13, 2, F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);