Finite Group Databasebeta

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#72226554

order9216
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC28 ⋊ D32
derived#72218415 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72218415C22
FrattiniC22#72215889
FittingC28D32
socleC26D3×S4
LCSG ▷ #72218415
UCSC1

At least 11491 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has at least 11491 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 16
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 16 points:

(1,3)(2,6,8,7)(9,11,12,15)(10,16,13,14);(1,2)(3,6)(4,8)(5,7)(9,10)(11,16,12,13,15,14)

Polycyclic presentation:

2^2,3^2,2^8|c^a=c^2,c^b=c^2,d^a=c^2d,d^b=d^2,e^a=efg,e^b=jk,e^c=eij,e^d=f,f^a=ehi,f^b=il,f^c=fi,f^d=ef,g^a=efghi,g^b=hijl,g^c=fgh,g^d=gj,h^a=j,h^b=efgk,h^c=fgi,h^d=fhj,i^a=egj,i^b=fkl,i^c=f,i^d=j,j^a=h,j^b=ek,j^c=ef,j^d=ij,k^c=l,k^d=kl,l^a=kl,l^b=kl,l^c=kl,l^d=k

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

automorphism group ofH(3,𝔽4)

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