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#72285397

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

Aliases

No aliases for this group.

Splittings

Fitting#8933101 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72258118
commutator, cosocle#46996⋊C3C22
FrattiniC22×C4C2×#1020
Fitting#8933101D3
socleC23#87524
prime radical#72231804C2
hypocenter#73292D4
LCSG ▷ #46996⋊C3 ▷ #73292
UCSC1 ◁ C2

1764 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 48 points:

(1,4,8,17)(2,14)(5,41,26,42)(9,13,12,15)(10,18)(11,48,28,30)(16,40,22,43)(19,37,27,38)(21,25,23,24)(29,44,36,39)(31,32,35,46)(33,45,34,47);(1,33,23,2,5,32,8,11,41,14,25,44)(3,35,21,15,16,46,7,36,42,13,27,39)(4,29,37,18,19,48,17,31,43,10,22,45)(6,34,38,9,26,30,20,28,40,12,24,47)

Polycyclic presentation:

2^3,3,2^7|a^2=ij,b^2=c,c^2=ij,e^2=i,f^2=ijk,b^a=bc,c^a=cij,d^a=dk,d^b=d^2jk,d^c=djk,e^a=ejk,e^b=fgi,e^c=ejk,e^d=fj,f^a=fj,f^b=ehi,f^c=fj,f^d=efij,f^e=fij,g^a=gj,g^b=hjk,g^c=gjk,g^d=gh,g^e=gj,g^f=gjk,h^a=hjk,h^b=gj,h^c=hk,h^d=g,h^e=hk,h^f=hj,i^d=ijk,j^d=k,k^b=jk,k^d=jk

Conjugacy classes & character table

43 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, 1, F.9*F.10);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4^2*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*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);