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#72245222

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC1G
commutator#1705⋊C3C2×C4
FrattiniC2×#127C2×S4
Fitting#232157D3
socleC22#4626⋊C4
prime radical, cosocle#19745⋊C3C22
hypocenterC82⋊C3C4⋊C4
LCSG ▷ #1705⋊C3 ▷ C82⋊C3
UCSC1

331 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,55,11,24,20,31,30,17,33,23,41,9)(2,27,39,37,29,59,8,26,19,45,25,18)(3,32,62,50,38,28,22,10,63,21,64,5)(4,49,36,7,58,14,61,12,15,6,43,35)(13,47,48,42,60,54)(34,40,51,52,46,44)(53,57,56);(2,8,10,32)(3,16,18,56)(5,27)(7,30)(9,37,36,64)(11,33,14,35)(12,46,41,60)(13,50,52,39)(15,45,31,38)(17,54,58,48)(19,42,21,34)(20,40,49,47)(22,53,59,57)(23,61)(25,62)(26,29)(28,63)(43,51,55,44)

Polycyclic presentation:

2^4,3,2^6|a^2=c,b^2=c,f^2=h,g^2=i,h^2=j,i^2=k,b^a=bcd,e^a=e^2,f^a=fhjk,f^b=fijk,f^c=fj,f^d=fhj,f^e=ghjk,g^a=fgj,g^b=ghijk,g^c=gk,g^d=gik,g^e=fg,h^a=hj,h^b=hk,h^d=hj,h^e=ij,i^a=hi,i^b=ijk,i^d=ik,i^e=hi,j^e=k,k^a=jk,k^e=jk

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245429
maximal normal subgroup of#72245429

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