Polytope of Type {41,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {41,2}*164
if this polytope has a name.
Group : SmallGroup(164,4)
Rank : 3
Schlafli Type : {41,2}
Number of vertices, edges, etc : 41, 41, 2
Order of s₀s₁s₂ : 82
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {41,2,2} of size 328
   {41,2,3} of size 492
   {41,2,4} of size 656
   {41,2,5} of size 820
   {41,2,6} of size 984
   {41,2,7} of size 1148
   {41,2,8} of size 1312
   {41,2,9} of size 1476
   {41,2,10} of size 1640
   {41,2,11} of size 1804
   {41,2,12} of size 1968
Vertex Figure Of :
   {2,41,2} of size 328
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {82,2}*328
   3-fold covers : {123,2}*492
   4-fold covers : {164,2}*656, {82,4}*656
   5-fold covers : {205,2}*820
   6-fold covers : {82,6}*984, {246,2}*984
   7-fold covers : {287,2}*1148
   8-fold covers : {164,4}*1312, {82,8}*1312, {328,2}*1312
   9-fold covers : {369,2}*1476, {123,6}*1476
   10-fold covers : {82,10}*1640, {410,2}*1640
   11-fold covers : {451,2}*1804
   12-fold covers : {82,12}*1968, {164,6}*1968a, {492,2}*1968, {246,4}*1968a, {123,6}*1968, {123,4}*1968
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)
(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41);;
s1 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40);;
s2 := (42,43);;
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, 
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*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*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*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(43)!( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)
(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41);
s1 := Sym(43)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)
(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40);
s2 := Sym(43)!(42,43);
poly := sub<Sym(43)|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, 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*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*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*s1*s0*s1*s0*s1 >;

to this polytope