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#72271313

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC22C22×#1020
commutator, cosocle, hypocenter#543⋊C3C23
Fitting#62844D3
prime radical#62844⋊C3C2
hypercenterC22⋊C4#1020
LCSG ▷ #543⋊C3
UCSC1 ◁ C22 ◁ C22⋊C4

1802 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 1802 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^3,3,2^6|b^2=j,c^2=i,g^2=i,h^2=i,b^a=bi,c^a=cj,c^b=ci,d^b=d^2,e^b=ef,e^d=f,f^d=ef,g^b=gh,g^d=h,g^f=gj,h^b=hi,h^d=ghi,h^e=hj,h^g=hi

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72271319

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