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#72228035

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#13343
commutator, Frattini, cosocle#2297C23
prime radicalC2×#585D4
LCSG ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1232 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,9)(4,13)(5,8)(6,11)(7,15)(10,12)(14,16)(17,18,21,27,23,29,44,60)(19,30,33,59,35,61,69,49)(20,36,38,62,40,72,42,53)(22,28,47,57,46,26,43,58)(24,41,50,45,51,32,64,34)(25,39,55,77,56,76,54,74)(31,52,66,71,68,80,75,48)(37,73,70,78,65,63,67,79);(1,3,5,12)(2,6,8,16)(4,14,15,11)(7,10,13,9)(17,19,23,35)(18,24,29,51)(20,37,40,65)(21,41,44,32)(22,45,46,34)(25,52,56,80)(26,30,28,61)(27,59,60,49)(31,38,68,42)(33,43,69,47)(36,71,72,48)(39,75,76,66)(50,58,64,57)(53,79,62,78)(54,73,55,63)(67,77,70,74);(1,2)(3,10)(4,7)(5,8)(6,11)(9,12)(13,15)(14,16)(17,20,23,40)(18,25,29,56)(19,31,35,68)(21,42,44,38)(22,39,46,76)(24,48,51,71)(26,53,28,62)(27,55,60,54)(30,63,61,73)(32,37,41,65)(33,66,69,75)(34,70,45,67)(36,57,72,58)(43,77,47,74)(49,79,59,78)(50,80,64,52)

Polycyclic presentation:

2^10|a^2=f,b^2=hj,c^2=j,d^2=h,e^2=j,f^2=j,g^2=j,b^a=bd,c^a=cfghij,c^b=cdej,d^a=dg,d^b=dh,e^a=eh,e^b=ej,e^c=ehj,f^b=fghj,f^c=fi,f^d=fij,g^a=gi,g^b=gj,g^c=gij,g^d=gj,g^f=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228042, #72228047
maximal normal subgroup of#72228042

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