Finite Group Databasebeta

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#72341177

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2AGL(2,ℤ/4)
commutator, cosocle#58990⋊C3C22
FrattiniC24C2×#1020
Fitting#8108057D3
socleC23#87524
prime radical#72247728C2
hypocenter#2679⋊C3D4
LCSG ▷ #58990⋊C3 ▷ #2679⋊C3
UCSC1 ◁ C2

3563 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 3563 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 48

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

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

Polycyclic presentation:

2^3,3,2^7|b^2=ik,g^2=ij,h^2=ijk,b^a=bc,d^a=dj,d^b=d^2,e^b=f,e^d=f,f^b=e,f^d=ef,g^a=gk,g^b=eghi,g^c=gj,g^d=hjk,g^e=gi,g^f=gj,h^a=hjk,h^b=efh,h^c=hk,h^d=ghik,h^e=hijk,h^f=hij,h^g=hik,i^d=ij,j^b=jk,j^d=k,k^d=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341182
maximal normal subgroup of#72341182

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