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#72227730

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11715850
commutator, prime radical, cosocle, hypocenter#72227729C2
Frattini, hypercenterC23#87588
Fitting#8093479D3
LCSG ▷ #72227729
UCSC1 ◁ C22 ◁ C23

1956 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2,3,2^9|a^2=j,c^2=j,d^2=j,b^a=b^2,c^a=dj,c^b=d,d^a=cj,d^b=cdj,d^c=dj,e^a=ef,e^b=f,e^c=ei,e^d=ek,f^b=ef,f^c=fik,f^d=fi,g^a=h,g^b=h,g^e=gj,g^f=gj,h^a=g,h^b=gh,h^f=hj,i^a=ik

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227737
maximal normal subgroup of#72227737

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