Finite Group Databasebeta

← Prev (#72266493) Next (#72266495) →

#72266494

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266483 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC23#72209707
commutator, Fitting, cosocle, hypocenter#72266483C3
FrattiniC25×Q8C22⋊A4
LCSG ▷ #72266483
UCSC1 ◁ C23

30864 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 30864 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 128

Polycyclic presentation:

3,2^12|b^2=f,c^2=fgh,d^2=fjm,e^2=fhjlm,f^2=m,g^2=m,h^2=m,i^2=m,j^2=m,b^a=cgl,c^a=bcfgm,c^b=cfgm,d^a=egh,d^b=dj,d^c=dhim,e^a=defghjklm,e^b=eijklm,e^c=egjl,e^d=efghjklm,f^a=fghkl,f^c=fk,f^d=fkl,g^a=hk,g^b=gk,g^c=gklm,g^d=gkm,g^e=gkl,h^a=ghkl,h^b=hklm,h^c=hlm,h^d=hklm,h^e=hl,h^f=hm,h^g=hm,i^a=hikl,i^b=iklm,i^c=ikl,i^e=ilm,i^f=im,i^g=im,i^h=im,j^a=gjm,j^b=jklm,j^c=jl,j^d=jkl,j^e=jlm,j^f=jm,j^g=jm,j^i=jm,k^a=lm,k^c=km,k^d=km,k^e=km,l^a=kl,l^b=lm,l^e=lm

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266550

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