Part of the Atlas of Small Regular Polytopes

Polytope of Type {15,6}

Atlas Canonical Name {15,6}*960

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Overview

Group
SmallGroup(960,5762)
Rank
3
Schläfli Type
{15,6}
Vertices, edges, …
80, 240, 32
Order of s0s1s2
40
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

4-fold

5-fold

20-fold

40-fold

48-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1*s0*(s1*s2)^2*s1> of order 2

16 facets

40 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 4

8 facets

20 vertex figures

P/N, where N=<(s1*s0)^2*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 4

8 facets

20 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,65)(18,66)(19,68)(20,67)(21,76)(22,75)(23,73)(24,74)(25,71)(26,72)(27,70)(28,69)(29,78)(30,77)(31,79)(32,80)(33,49)(34,50)(35,52)(36,51)(37,60)(38,59)(39,57)(40,58)(41,55)(42,56)(43,54)(44,53)(45,62)(46,61)(47,63)(48,64);;
s1 := ( 1,17)( 2,19)( 3,18)( 4,20)( 5,24)( 6,22)( 7,23)( 8,21)( 9,32)(10,30)(11,31)(12,29)(13,28)(14,26)(15,27)(16,25)(33,65)(34,67)(35,66)(36,68)(37,72)(38,70)(39,71)(40,69)(41,80)(42,78)(43,79)(44,77)(45,76)(46,74)(47,75)(48,73)(50,51)(53,56)(57,64)(58,62)(59,63)(60,61);;
s2 := ( 1,15)( 2,16)( 3,14)( 4,13)( 5, 6)(11,12)(17,31)(18,32)(19,30)(20,29)(21,22)(27,28)(33,47)(34,48)(35,46)(36,45)(37,38)(43,44)(49,63)(50,64)(51,62)(52,61)(53,54)(59,60)(65,79)(66,80)(67,78)(68,77)(69,70)(75,76);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(80)!( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,65)(18,66)(19,68)(20,67)(21,76)(22,75)(23,73)(24,74)(25,71)(26,72)(27,70)(28,69)(29,78)(30,77)(31,79)(32,80)(33,49)(34,50)(35,52)(36,51)(37,60)(38,59)(39,57)(40,58)(41,55)(42,56)(43,54)(44,53)(45,62)(46,61)(47,63)(48,64);
s1 := Sym(80)!( 1,17)( 2,19)( 3,18)( 4,20)( 5,24)( 6,22)( 7,23)( 8,21)( 9,32)(10,30)(11,31)(12,29)(13,28)(14,26)(15,27)(16,25)(33,65)(34,67)(35,66)(36,68)(37,72)(38,70)(39,71)(40,69)(41,80)(42,78)(43,79)(44,77)(45,76)(46,74)(47,75)(48,73)(50,51)(53,56)(57,64)(58,62)(59,63)(60,61);
s2 := Sym(80)!( 1,15)( 2,16)( 3,14)( 4,13)( 5, 6)(11,12)(17,31)(18,32)(19,30)(20,29)(21,22)(27,28)(33,47)(34,48)(35,46)(36,45)(37,38)(43,44)(49,63)(50,64)(51,62)(52,61)(53,54)(59,60)(65,79)(66,80)(67,78)(68,77)(69,70)(75,76);
poly := sub<Sym(80)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2 >;

References

None.

to this polytope.

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