Finite Group Databasebeta

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#72341170

order12288
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#72340983 ⋊ D3
derived#72341029 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72341029C2
FrattiniC25#87588
Fitting#72340983D3
socleC22#72340869
LCSG ▷ #72341029
UCSC1

5776 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 5776 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

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,48,22,35,8,64,7,15)(2,57,52,21,17,60,50,41)(3,45,5,38,39,19,20,47)(4,18,23,32)(6,11,40,36,43,12,55,16)(9,42,49,30,10,63,44,27)(13,59,24,25,37,31,26,56)(28,61,62,46)(29,54,58,53)(33,51);(1,8,9)(2,12,11)(3,32,30)(4,15,24)(5,33,43)(6,13,51)(14,21,37)(16,50,36)(18,27,26)(19,61,40)(20,34,41)(22,49,44)(23,35,39)(25,29,57)(28,42,38)(31,54,63)(45,46,60)(47,53,48)(55,56,58)(59,62,64);(1,9)(2,11)(3,59,24,38)(4,33,32,14)(5,56,37,45)(6,48,41,63)(7,22)(8,10)(12,17)(13,19,20,25)(15,55,30,60)(18,34,23,51)(21,42,43,64)(26,47,39,31)(27,57,35,40)(28,46,62,58)(29,53,61,54)(44,49)

Polycyclic presentation:

2,3,2^11|a^2=lm,c^2=i,d^2=i,e^2=ikm,f^2=ikm,g^2=jkl,h^2=jkl,b^a=b^2l,c^a=cilm,c^b=d,d^a=cdl,d^b=cdi,d^c=di,e^a=fikm,e^b=f,e^c=eij,e^d=ek,f^a=eikm,f^b=efikm,f^c=fijk,f^d=fijl,f^e=fikm,g^a=ehm,g^b=gh,g^c=gijlm,g^d=gkm,g^e=gjm,g^f=gim,h^a=fgl,h^b=g,h^c=hkm,h^d=hijk,h^e=hilm,h^f=hijl,h^g=hjkl,i^e=im,i^f=il,i^g=im,i^h=ilm,j^a=jkm,j^b=jl,j^c=jm,j^d=jl,j^e=jl,j^f=jlm,j^g=jlm,j^h=jl,k^a=klm,k^b=klm,k^c=klm,k^d=km,k^e=km,k^f=kl,k^g=klm,k^h=kl,l^a=m,l^b=lm,m^a=l,m^b=l

Conjugacy classes & character table

51 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.12*F.13);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9*F.11*F.13);;
gap> SetPower(coll, 6, F.9*F.11*F.13);;
gap> SetPower(coll, 7, F.10*F.11*F.12);;
gap> SetPower(coll, 8, F.10*F.11*F.12);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.12*F.13);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.4*F.12);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.6*F.9*F.11*F.13);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.5*F.9*F.11*F.13);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.9*F.11*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.11*F.13);;
gap> SetConjugate(coll, 7, 1, F.5*F.8*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.11*F.13);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.13);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.13);;
gap> SetConjugate(coll, 8, 1, F.6*F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.11*F.13);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.12*F.13);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.10*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.13);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 9, 7, F.9*F.13);;
gap> SetConjugate(coll, 9, 8, F.9*F.12*F.13);;
gap> SetConjugate(coll, 10, 1, F.10*F.11*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.13);;
gap> SetConjugate(coll, 10, 4, F.10*F.12);;
gap> SetConjugate(coll, 10, 5, F.10*F.12);;
gap> SetConjugate(coll, 10, 6, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 7, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 8, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 3, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 4, F.11*F.13);;
gap> SetConjugate(coll, 11, 5, F.11*F.13);;
gap> SetConjugate(coll, 11, 6, F.11*F.12);;
gap> SetConjugate(coll, 11, 7, F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 8, F.11*F.12);;
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);