Polytope of Type {6,123}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope