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#72235274

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C42

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72235251 ⋊ C3
derived#346281 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#346281C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72235251C3
socleC22C2×#73292
LCSG ▷ #346281
UCSC1

1098 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 28 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=gi,e^2=jk,f^2=j,b^a=bhi,c^b=dhjk,d^a=djk,d^b=cdgh,d^c=dgh,e^a=ejk,e^b=fjk,e^c=ehik,e^d=ehjk,f^a=fj,f^b=ef,f^c=fhk,f^d=fik,g^b=gij,g^d=gk,g^e=gj,g^f=gk,h^b=hik,h^c=hk,h^d=hj,i^b=hjk,i^c=ijk,i^d=ik,j^b=k,k^b=jk

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235276

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