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#72227679

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC22#72209647
commutator, prime radical, cosocle#72227668C22
FrattiniC23×C42C2×S4
Fitting#72227647D3
socleC24#72991
hypocenter#17072⋊C3D4
LCSG ▷ #72227668 ▷ #17072⋊C3
UCSC1 ◁ C22

2406 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 56
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 2 generators on 56 points:

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

Polycyclic presentation:

2^3,3,2^8|a^2=c,b^2=il,e^2=i,f^2=ij,g^2=k,h^2=j,b^a=bckl,d^b=d^2,e^a=egjkl,e^b=efhjkl,e^c=ek,e^d=fj,f^a=fghl,f^b=fghikl,f^c=fjk,f^d=efi,f^e=fik,g^b=hjk,g^d=gh,g^e=gk,g^f=gjkl,h^b=gjk,h^d=gj,h^e=hjl,h^f=hl,i^b=ijk,i^d=ij,j^b=k,j^d=k,k^b=j,k^d=jk,l^b=jkl,l^d=jl

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227690

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