Finite Group Databasebeta

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#72344160

order12288
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Splittings

Fitting#72241349 ⋊ D3
derived#72340920 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72340920C2
FrattiniC25#87588
Fitting#72241349D3
socleC22#72227766
LCSG ▷ #72340920
UCSC1

Only the 41 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 41 normal subgroups are shown; non-normal subgroups are omitted.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^11|e^2=i,f^2=i,g^2=l,h^2=lm,b^a=b^2,c^b=d,d^a=cd,d^b=cd,e^a=ce,e^b=f,e^d=ei,f^a=cdefi,f^b=efi,f^c=fi,f^e=fi,g^a=ghlm,g^b=hl,g^c=gk,g^d=gkl,g^e=gij,g^f=gijkm,h^a=hlm,h^b=gh,h^c=hlm,h^d=hklm,h^e=hklm,h^f=hijlm,i^g=il,i^h=ilm,j^a=jlm,j^b=jlm,j^c=jlm,j^d=jm,j^e=jl,j^f=jlm,j^g=jl,j^h=jlm,k^a=klm,k^b=klm,k^e=klm,k^f=km,l^a=m,l^b=lm,m^a=l,m^b=l

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344184
maximal normal subgroup of#72344184

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