Finite Group Databasebeta

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#72268171

order12288
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268055 ⋊ C3
derived#72268050 ⋊ C2×C3

Subgroups

subgroupG/H
centerC2#72227398
commutator, cosocle, hypocenter#72268050C2×C3
FrattiniC25×#41C2×A4
Fitting#72268055C3
socleC23#72209914
hypercenterC22#72227397
LCSG ▷ #72268050
UCSC1 ◁ C2 ◁ C22

Only the 40 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 40 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|c^2=g,d^2=h,e^2=m,f^2=m,g^2=m,h^2=m,i^2=m,j^2=m,b^a=bkl,c^a=ce,c^b=cd,d^a=df,d^b=chikm,d^c=di,e^a=em,e^b=efijm,e^d=eijm,f^a=fm,f^b=e,f^c=fijm,f^e=fm,g^a=gm,g^b=ghilm,g^d=gk,h^a=hm,h^b=gklm,h^c=hl,i^a=im,i^b=ikm,i^c=ikm,i^d=ilm,j^b=jkm,j^c=jkm,j^d=jlm,k^b=klm,l^b=k

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268920

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