Polytope of Type {40,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40,4}*320b
if this polytope has a name.
Group : SmallGroup(320,374)
Rank : 3
Schlafli Type : {40,4}
Number of vertices, edges, etc : 40, 80, 4
Order of s₀s₁s₂ : 40
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {40,4,2} of size 640
   {40,4,4} of size 1280
   {40,4,6} of size 1920
Vertex Figure Of :
   {2,40,4} of size 640
   {4,40,4} of size 1280
   {4,40,4} of size 1280
   {6,40,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,4}*160
   4-fold quotients : {20,2}*80, {10,4}*80
   5-fold quotients : {8,4}*64b
   8-fold quotients : {10,2}*40
   10-fold quotients : {4,4}*32
   16-fold quotients : {5,2}*20
   20-fold quotients : {2,4}*16, {4,2}*16
   40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {40,4}*640a, {40,8}*640c, {40,8}*640d
   3-fold covers : {40,12}*960b, {120,4}*960b
   4-fold covers : {40,8}*1280a, {40,4}*1280a, {40,8}*1280d, {80,4}*1280a, {80,4}*1280b, {80,8}*1280a, {80,8}*1280b, {40,16}*1280c, {40,16}*1280e
   5-fold covers : {200,4}*1600b, {40,20}*1600e, {40,20}*1600f
   6-fold covers : {120,4}*1920a, {40,12}*1920a, {120,8}*1920a, {40,24}*1920b, {120,8}*1920d, {40,24}*1920d
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {40,4}

Twisty Puzzle