Polytope of Type {2,162}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,162}*648
if this polytope has a name.
Group : SmallGroup(648,43)
Rank : 3
Schlafli Type : {2,162}
Number of vertices, edges, etc : 2, 162, 162
Order of s₀s₁s₂ : 162
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,162,2} of size 1296
Vertex Figure Of :
   {2,2,162} of size 1296
   {3,2,162} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,81}*324
   3-fold quotients : {2,54}*216
   6-fold quotients : {2,27}*108
   9-fold quotients : {2,18}*72
   18-fold quotients : {2,9}*36
   27-fold quotients : {2,6}*24
   54-fold quotients : {2,3}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,324}*1296, {4,162}*1296a
   3-fold covers : {2,486}*1944, {6,162}*1944a, {6,162}*1944b
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6, 10)(  7,  9)(  8, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 21)
( 16, 23)( 17, 22)( 18, 28)( 19, 27)( 20, 29)( 30, 66)( 31, 68)( 32, 67)
( 33, 73)( 34, 72)( 35, 74)( 36, 70)( 37, 69)( 38, 71)( 39, 57)( 40, 59)
( 41, 58)( 42, 64)( 43, 63)( 44, 65)( 45, 61)( 46, 60)( 47, 62)( 48, 78)
( 49, 80)( 50, 79)( 51, 75)( 52, 77)( 53, 76)( 54, 82)( 55, 81)( 56, 83)
( 85, 86)( 87, 91)( 88, 90)( 89, 92)( 93,105)( 94,107)( 95,106)( 96,102)
( 97,104)( 98,103)( 99,109)(100,108)(101,110)(111,147)(112,149)(113,148)
(114,154)(115,153)(116,155)(117,151)(118,150)(119,152)(120,138)(121,140)
(122,139)(123,145)(124,144)(125,146)(126,142)(127,141)(128,143)(129,159)
(130,161)(131,160)(132,156)(133,158)(134,157)(135,163)(136,162)(137,164);;
s2 := (  3,111)(  4,113)(  5,112)(  6,118)(  7,117)(  8,119)(  9,115)( 10,114)
( 11,116)( 12,132)( 13,134)( 14,133)( 15,129)( 16,131)( 17,130)( 18,136)
( 19,135)( 20,137)( 21,123)( 22,125)( 23,124)( 24,120)( 25,122)( 26,121)
( 27,127)( 28,126)( 29,128)( 30, 84)( 31, 86)( 32, 85)( 33, 91)( 34, 90)
( 35, 92)( 36, 88)( 37, 87)( 38, 89)( 39,105)( 40,107)( 41,106)( 42,102)
( 43,104)( 44,103)( 45,109)( 46,108)( 47,110)( 48, 96)( 49, 98)( 50, 97)
( 51, 93)( 52, 95)( 53, 94)( 54,100)( 55, 99)( 56,101)( 57,147)( 58,149)
( 59,148)( 60,154)( 61,153)( 62,155)( 63,151)( 64,150)( 65,152)( 66,138)
( 67,140)( 68,139)( 69,145)( 70,144)( 71,146)( 72,142)( 73,141)( 74,143)
( 75,159)( 76,161)( 77,160)( 78,156)( 79,158)( 80,157)( 81,163)( 82,162)
( 83,164);;
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*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(164)!(1,2);
s1 := Sym(164)!(  4,  5)(  6, 10)(  7,  9)(  8, 11)( 12, 24)( 13, 26)( 14, 25)
( 15, 21)( 16, 23)( 17, 22)( 18, 28)( 19, 27)( 20, 29)( 30, 66)( 31, 68)
( 32, 67)( 33, 73)( 34, 72)( 35, 74)( 36, 70)( 37, 69)( 38, 71)( 39, 57)
( 40, 59)( 41, 58)( 42, 64)( 43, 63)( 44, 65)( 45, 61)( 46, 60)( 47, 62)
( 48, 78)( 49, 80)( 50, 79)( 51, 75)( 52, 77)( 53, 76)( 54, 82)( 55, 81)
( 56, 83)( 85, 86)( 87, 91)( 88, 90)( 89, 92)( 93,105)( 94,107)( 95,106)
( 96,102)( 97,104)( 98,103)( 99,109)(100,108)(101,110)(111,147)(112,149)
(113,148)(114,154)(115,153)(116,155)(117,151)(118,150)(119,152)(120,138)
(121,140)(122,139)(123,145)(124,144)(125,146)(126,142)(127,141)(128,143)
(129,159)(130,161)(131,160)(132,156)(133,158)(134,157)(135,163)(136,162)
(137,164);
s2 := Sym(164)!(  3,111)(  4,113)(  5,112)(  6,118)(  7,117)(  8,119)(  9,115)
( 10,114)( 11,116)( 12,132)( 13,134)( 14,133)( 15,129)( 16,131)( 17,130)
( 18,136)( 19,135)( 20,137)( 21,123)( 22,125)( 23,124)( 24,120)( 25,122)
( 26,121)( 27,127)( 28,126)( 29,128)( 30, 84)( 31, 86)( 32, 85)( 33, 91)
( 34, 90)( 35, 92)( 36, 88)( 37, 87)( 38, 89)( 39,105)( 40,107)( 41,106)
( 42,102)( 43,104)( 44,103)( 45,109)( 46,108)( 47,110)( 48, 96)( 49, 98)
( 50, 97)( 51, 93)( 52, 95)( 53, 94)( 54,100)( 55, 99)( 56,101)( 57,147)
( 58,149)( 59,148)( 60,154)( 61,153)( 62,155)( 63,151)( 64,150)( 65,152)
( 66,138)( 67,140)( 68,139)( 69,145)( 70,144)( 71,146)( 72,142)( 73,141)
( 74,143)( 75,159)( 76,161)( 77,160)( 78,156)( 79,158)( 80,157)( 81,163)
( 82,162)( 83,164);
poly := sub<Sym(164)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

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