Finite Group Databasebeta

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#72271486

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72271474 ⋊ C3
derived#72268378 ⋊ C22×C3

Subgroups

subgroupG/H
center, socleC22#72209720
commutator, cosocle, hypocenter#72268378C22×C3
FrattiniC22×Q82C22×A4
Fitting#72271474C3
hypercenterC24#11709314
LCSG ▷ #72268378
UCSC1 ◁ C22 ◁ C24

At least 13471 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has at least 13471 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 32
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,5,9)(3,16,15,6,11,14,12,7,4,10,13,8)(17,18,19,24)(20,26,31,25,21,32,27,22,30,29,28,23);(2,8)(4,15)(6,7)(9,14)(10,16)(11,13)(17,25,20,18,19,29,27,24)(21,23,31,22,28,32,30,26)

Polycyclic presentation:

2^2,3,2^10|a^2=lm,b^2=l,d^2=j,e^2=j,f^2=l,g^2=l,h^2=lm,i^2=lm,d^a=dh,d^b=dg,d^c=de,e^a=ehilm,e^b=ef,e^c=d,e^d=ej,f^a=flm,f^b=fl,f^c=g,f^d=fk,f^e=fm,g^a=glm,g^b=gl,g^c=fgklm,g^d=gm,g^e=gkm,g^f=gl,h^a=hlm,h^b=hlm,h^c=ijlm,h^e=hjlm,h^f=hlm,i^c=hij,i^d=ijlm,i^e=ijlm,i^f=ilm,i^g=ilm,i^h=ilm,j^a=jlm,j^b=jlm,k^a=klm,k^b=kl

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271711

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