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#72227673

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC22#85558
commutator, prime radical, cosocle, hypocenter#17072⋊C3C2
Frattini, socleC24#1020
Fitting#17072D3
LCSG ▷ #17072⋊C3
UCSC1 ◁ C22

436 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 17 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) = 32
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 32 points:

(1,5,3,6)(2,4,8,7)(9,18)(10,14)(11,28,21,29,22,31,23,19)(12,16,13,17)(15,24,25,30)(20,26,32,27);(2,3,5)(6,7,8)(9,11,20)(10,15,19)(12,23,30)(13,21,32)(14,22,24)(16,25,31)(17,26,28)(18,27,29)

Polycyclic presentation:

2,3,2^8|a^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=h,b^a=b^2,c^a=cdhi,c^b=di,d^a=dgh,d^b=cdgi,d^c=dgi,e^a=ehi,e^b=fhi,e^c=ehij,e^d=ehi,f^a=ef,f^b=ef,f^c=fhj,f^d=fj,g^a=ghi,g^b=gh,h^a=i,h^b=i,i^a=h,i^b=hi,j^a=hij,j^b=hj

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227674, #72266667
maximal normal subgroup of#72227674, #72266667
maximal quotient of#72266674

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