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#72340897

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

Aliases

No aliases for this group.

Splittings

Fitting#72228245 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72227467C22
FrattiniC25C2×#1020
Fitting#72228245D3
socleC22#72227614
LCSG ▷ #72227467
UCSC1

1760 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 1760 conjugacy classes of subgroups — too many to draw. Only its 30 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:

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

Polycyclic presentation:

2^2,3,2^9|b^2=l,d^2=i,e^2=i,f^2=ijkl,g^2=ijkl,b^a=bijk,c^a=ckl,c^b=c^2k,d^a=dk,d^b=dil,d^c=e,e^a=el,e^b=dek,e^c=dei,e^d=ei,f^b=fgkl,f^c=g,f^d=fh,f^e=fhj,g^b=gijk,g^c=fgijkl,g^d=gjkl,g^e=ghk,g^f=gijkl,h^b=hjkl,h^c=hk,h^d=hk,h^e=hl,h^f=hl,h^g=hkl,i^f=ik,i^g=il,j^b=jl,j^c=jl,j^d=jl,j^e=jkl,j^f=jk,j^g=jl,k^b=kl,k^c=l,l^c=kl

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