Finite Group Databasebeta

← Prev (#72220495) Next (#72220497) →

#72220496

order31104
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, socleC34 ⋊ #85555
derived#72220482 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenter#72220482C2
Fitting, socleC34#85555
LCSG ▷ #72220482
UCSC1

886 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^6,3^4|c^2=h,d^2=h,e^2=g,f^2=g,b^a=b^2,c^a=cd,c^b=d,d^a=dh,d^b=cdh,d^c=dh,e^a=ef,e^b=f,e^d=eh,f^a=fg,f^b=efg,f^c=fh,f^e=fg,i^a=ik^2,i^b=ik,i^c=k^2,i^d=i^2k,i^e=k^2,i^f=i^2k,i^g=i^2,i^h=i^2,j^a=jk^2l^2,j^b=jk^2l,j^c=jk,j^d=i^2jk,j^e=ikl^2,j^f=ij^2kl,j^g=j^2,j^h=ijk,k^a=k^2,k^c=i,k^d=ik,k^e=i,k^f=ik,k^g=k^2,k^h=k^2,l^a=l^2,l^c=i^2kl,l^d=i^2l,l^e=ijk^2,l^f=ijk^2l,l^g=l^2,l^h=k^2l

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

holomorph ofPSU(3,𝔽2)

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