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#72267842

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting#72267796 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC22#72219762
commutator, cosocle#72267792C22
FrattiniC25×C4C2×S4
Fitting#72267796D3
socleC26GL(2,ℤ/4)
prime radical#72267840C2
hypocenterC22⋊(#95⋊C3)SD16
LCSG ▷ #72267792 ▷ C2×C22⋊(#95⋊C3) ▷ C22⋊(#95⋊C3)
UCSC1 ◁ C22

5676 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 5676 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 40
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^4,3,2^7|a^2=dl,c^2=d,f^2=j,g^2=k,b^a=bc,c^a=cd,c^b=cd,e^a=ejl,e^b=e^2,e^c=ekl,f^a=fhi,f^b=gj,f^c=fjl,f^e=gjl,g^a=gh,g^b=fk,g^c=gkl,g^e=fgl,g^f=gl,h^b=hijl,h^e=i,i^b=ijk,i^e=hi,j^b=k,j^e=k,k^b=j,k^e=jkl

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267843

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4*F.12);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.12);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.12);;
gap> SetConjugate(coll, 6, 5, F.7*F.10*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10*F.12);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> SetConjugate(coll, 11, 5, F.10*F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);