Finite Group Databasebeta

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#72258671

order9216
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC28 ⋊ D32
derivedC22⋊A42 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊A42C22
Frattini, socleC24S42
FittingC28D32
LCSG ▷ C22⋊A42
UCSC1

18088 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 18088 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 24

Polycyclic presentation:

2^2,3^2,2^8|c^a=cd,c^b=c^2,d^a=d^2,d^b=d^2,e^a=efjl,e^b=ef,e^c=f,e^d=fl,f^a=fil,f^c=ef,f^d=efkl,g^a=gk,g^c=h,h^a=hl,h^b=gh,h^c=gh,i^b=ik,i^c=j,i^d=jk,j^a=ijl,j^b=ijkl,j^c=ij,j^d=ijl,k^c=l,l^b=kl,l^c=kl

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258869, #72258916
maximal normal subgroup of#72258869, #72258916

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