Finite Group Databasebeta

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#72268534

order12288
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72235024 ⋊ C3
derived#72235019 ⋊ C22×C3

Subgroups

subgroupG/H
center, socleC22#72209720
commutator, cosocle, hypocenter#72235019C22×C3
FrattiniC22×H(3,𝔽4)C22×A4
Fitting#72235024C3
hypercenterC24#11709314
LCSG ▷ #72235019
UCSC1 ◁ C22 ◁ C24

41767 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 41767 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 64

Polycyclic presentation:

2^2,3,2^10|b^2=m,f^2=l,g^2=l,h^2=m,i^2=m,b^a=blm,d^a=df,d^b=dhim,d^c=e,e^a=eg,e^b=eh,e^c=de,f^a=fl,f^b=flm,f^c=g,f^d=fl,f^e=fj,g^a=gl,g^b=glm,g^c=fgj,g^d=gjl,g^e=gl,g^f=gl,h^a=hlm,h^b=hm,h^c=ik,h^d=hk,h^e=hm,h^f=hlm,i^c=hikm,i^d=ikm,i^e=ik,i^f=ilm,i^g=ilm,i^h=im,j^a=jl,j^b=jlm,k^a=klm,k^b=km

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269016

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