Finite Group Databasebeta

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#72239537

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72239526 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209996
commutator, Fitting, cosocle, hypocenter#72239526C3
FrattiniC22×C43C22⋊A4
socleC23#72209995
LCSG ▷ #72239526
UCSC1 ◁ C2

11733 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 11733 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,7)(3,12,17)(4,11,23)(5,8,18)(6,10,19)(9,20,15)(13,22,16)(14,21,24);(3,15,14,13)(9,21,22,12)(16,20)(17,24);(1,13)(2,12)(3,4)(5,14)(6,15)(7,23,19,18)(8,22)(9,11)(10,21)(16,24,20,17)

Polycyclic presentation:

3,2^12|b^2=klm,c^2=l,d^2=f,e^2=g,h^2=k,i^2=l,j^2=m,b^a=ck,c^a=bclm,c^b=clm,d^a=ef,d^b=dh,d^c=dhil,e^a=de,e^b=ehilm,e^c=ej,f^a=g,f^b=fk,f^c=fkl,g^a=fg,g^b=gkl,g^c=gm,h^a=jkl,h^b=hk,h^c=hkl,i^a=hiklm,i^b=il,j^a=hjklm,j^b=jkl,j^c=jm,k^a=m,l^a=kl,m^a=km

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239549

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