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#72228032

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC23#87588
commutator, prime radical, cosocle, hypocenter#72227697C2
Fitting#8930496D3
LCSG ▷ #72227697
UCSC1 ◁ C23

2267 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 2267 conjugacy classes of subgroups — too many to draw. Only its 36 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 — 3 generators on 80 points:

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

Polycyclic presentation:

2,3,2^9|a^2=k,c^2=i,d^2=i,e^2=k,f^2=k,g^2=k,h^2=k,b^a=b^2,c^a=cghjk,c^b=d,d^a=cdgijk,d^b=cdi,d^c=di,e^a=defghijk,e^b=f,e^c=ei,f^a=cfgij,f^b=efk,f^c=fi,f^d=fi,f^e=fk,g^a=hk,g^b=gh,g^c=gij,g^d=gijk,g^e=gij,g^f=gijk,h^a=gk,h^b=g,h^c=hijk,h^d=hk,h^e=hijk,h^f=hk,h^g=hk

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228046
maximal normal subgroup of#72228046

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