Finite Group Databasebeta

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#72341247

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#59192⋊C3C22
FrattiniC24C2×#1020
Fitting#8108058D3
socleC23#87524
prime radical#72247734C2
hypocenter#2736⋊C3D4
LCSG ▷ #59192⋊C3 ▷ #2736⋊C3
UCSC1 ◁ C2

3450 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 3450 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:

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

Polycyclic presentation:

2^3,3,2^7|b^2=ij,g^2=ij,h^2=ij,b^a=bc,d^a=dj,d^b=d^2,d^c=djk,e^b=f,e^d=ef,f^b=e,f^d=e,g^a=gjk,g^b=hik,g^c=gk,g^d=h,g^e=gi,g^f=gk,h^a=hj,h^b=gijk,h^c=hjk,h^d=ghij,h^e=hjk,h^f=hi,h^g=hij,i^d=ijk,j^d=k,k^b=jk,k^d=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

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