Polytope of Type {123,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {123,6}*1968
if this polytope has a name.
Group : SmallGroup(1968,188)
Rank : 3
Schlafli Type : {123,6}
Number of vertices, edges, etc : 164, 492, 8
Order of s₀s₁s₂ : 164
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   12-fold quotients : {41,2}*164
   41-fold quotients : {3,6}*48
   82-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope