Finite Group Databasebeta

← Prev (#72260831) Next (#72260833) →

#72260832

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#8058967 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2AGL(2,ℤ/4)
commutator, cosocle#47814⋊C3C22
FrattiniC24C2×#1020
Fitting#8058967D3
socleC23#87524
hypocenter#2844⋊C3D4
LCSG ▷ #47814⋊C3 ▷ #2844⋊C3
UCSC1 ◁ C2

2038 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 2038 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) = 32
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,16,7,11,6,15,13)(3,5,8,14,9,4,10,12)(17,28,20,26,19,27,25,21)(18,31,23,24,22,32,29,30);(2,7,16,6,9,14)(3,15,10,13,12,8)(4,5,11)(17,18,19,22)(20,24,27,29,21,32,25,30,28,23,26,31)

Polycyclic presentation:

2^3,3,2^7|a^2=i,b^2=ci,e^2=i,f^2=i,g^2=ijk,h^2=ij,b^a=bc,d^b=d^2,e^b=ei,e^d=f,f^b=ef,f^d=efi,f^e=fi,g^a=gjk,g^b=efh,g^c=gj,g^d=hjk,g^e=gk,g^f=gij,h^a=hj,h^b=fgij,h^c=hk,h^d=ghi,h^e=hij,h^f=hjk,h^g=hi,j^b=k,j^d=k,k^b=j,k^d=jk

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72260895
maximal normal subgroup of#72260895
maximal quotient of#72261895

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