Finite Group Databasebeta

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#72267786

order12288
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267784 ⋊ C3
derived#72267782 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2#72226653
commutator, cosocle, hypocenter#72267782C2×C3
FrattiniC2×#7555C2×A4
Fitting#72267784C3
socleC23#72226651
LCSG ▷ #72267782
UCSC1 ◁ C2

11532 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 11532 conjugacy classes of subgroups — too many to draw. Only its 27 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=km,c^2=h,d^2=g,e^2=gjk,f^2=hjk,b^a=bhj,c^a=cf,c^b=cd,d^a=de,d^b=cgjm,d^c=dj,e^a=egjkl,e^b=fm,e^c=egi,e^d=egjm,f^a=fhjkm,f^b=efgikm,f^c=fhjm,f^d=fgilm,f^e=fkl,g^a=gkm,g^b=hlm,g^c=gl,g^e=gl,g^f=gm,h^a=hkm,h^b=ghjl,h^d=hm,h^e=hl,h^f=hm,i^b=gijl,i^d=im,j^a=jkm,j^b=jm,j^c=jm,j^d=jl,j^e=jl,j^f=jm,k^b=kl,k^c=km,k^d=kl,l^b=m,m^b=lm

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267787

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