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#72245125

order3072
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72245124 ⋊ C3
derivedC82 ⋊ C22×C3×C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC82C22×C3×C4
FrattiniC2×C42C23×A4
Fitting#72245124C3
socleC22C2×#2380⋊C3
prime radical#72245025C2
cosocle#1455C23×C3
LCSG ▷ C82
UCSC1

985 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 87 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:

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

Polycyclic presentation:

2^3,3,2^7|a^2=e,f^2=h,g^2=i,h^2=j,i^2=k,f^a=fi,f^b=fjk,f^c=fhj,f^d=ghjk,f^e=fj,g^a=ghik,g^b=gj,g^c=gik,g^d=fg,g^e=gk,h^a=hk,h^c=hj,h^d=ij,i^a=ijk,i^c=ik,i^d=hi,j^d=k,k^d=jk

Conjugacy classes & character table

69 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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
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, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 4, F.11);;
gap> SetConjugate(coll, 11, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);