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#72228047

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11714279
commutator, prime radical, cosocle, hypocenter#72209625C2
FrattiniC23×C4#1020
Fitting#432837D3
hypercenterC2×C4#85559
LCSG ▷ #72209625
UCSC1 ◁ C22 ◁ C2×C4

828 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 15 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) = 80
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 2 generators on 80 points:

(1,10,4,8)(2,14,15,5)(3,9,11,16)(6,13,7,12)(17,65,43,40,24,71,69,20)(18,70,60,59,29,77,75,26)(19,80,45,74,33,57,44,49)(21,78,32,61,35,56,47,64)(22,72,31,48,23,34,55,39)(25,38,63,67,53,76,62,30)(27,37,51,42,41,66,52,73)(28,79,50,58,46,54,68,36);(3,4,13)(6,7,15)(8,9,16)(11,12,14)(18,19,25)(20,35,73)(21,42,40)(26,55,79)(27,34,49)(28,47,51)(29,33,53)(30,50,61)(31,54,59)(32,52,46)(36,69,77)(37,75,80)(38,48,45)(39,44,76)(41,72,74)(43,70,58)(56,63,71)(57,66,60)(62,65,78)(64,67,68)

Polycyclic presentation:

2,3,2^9|a^2=j,c^2=j,d^2=j,e^2=gj,f^2=gijk,g^2=k,h^2=k,b^a=b^2,c^a=cj,c^b=d,d^a=cd,d^b=cdj,d^c=dj,e^a=defgijk,e^b=fghik,e^c=eg,e^d=eh,f^a=cfik,f^b=efhij,f^c=fi,f^d=fgi,f^e=fhij,g^a=gij,g^b=gi,g^c=gk,g^f=gk,h^a=hj,h^b=g,h^d=hk,h^e=hk,h^f=hk,i^b=ghik,i^d=ik,i^e=ik,i^f=ik

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228054
maximal normal subgroup of#72228054

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