Polytope of Type {4,123,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,123,2}*1968
if this polytope has a name.
Group : SmallGroup(1968,193)
Rank : 4
Schlafli Type : {4,123,2}
Number of vertices, edges, etc : 4, 246, 123, 2
Order of s₀s₁s₂s₃ : 246
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   41-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164);;
s1 := (  2,  3)(  5,161)(  6,163)(  7,162)(  8,164)(  9,157)( 10,159)( 11,158)
( 12,160)( 13,153)( 14,155)( 15,154)( 16,156)( 17,149)( 18,151)( 19,150)
( 20,152)( 21,145)( 22,147)( 23,146)( 24,148)( 25,141)( 26,143)( 27,142)
( 28,144)( 29,137)( 30,139)( 31,138)( 32,140)( 33,133)( 34,135)( 35,134)
( 36,136)( 37,129)( 38,131)( 39,130)( 40,132)( 41,125)( 42,127)( 43,126)
( 44,128)( 45,121)( 46,123)( 47,122)( 48,124)( 49,117)( 50,119)( 51,118)
( 52,120)( 53,113)( 54,115)( 55,114)( 56,116)( 57,109)( 58,111)( 59,110)
( 60,112)( 61,105)( 62,107)( 63,106)( 64,108)( 65,101)( 66,103)( 67,102)
( 68,104)( 69, 97)( 70, 99)( 71, 98)( 72,100)( 73, 93)( 74, 95)( 75, 94)
( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)( 81, 85)( 82, 87)( 83, 86)
( 84, 88);;
s2 := (  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);;
s3 := (165,166);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(166)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164);
s1 := Sym(166)!(  2,  3)(  5,161)(  6,163)(  7,162)(  8,164)(  9,157)( 10,159)
( 11,158)( 12,160)( 13,153)( 14,155)( 15,154)( 16,156)( 17,149)( 18,151)
( 19,150)( 20,152)( 21,145)( 22,147)( 23,146)( 24,148)( 25,141)( 26,143)
( 27,142)( 28,144)( 29,137)( 30,139)( 31,138)( 32,140)( 33,133)( 34,135)
( 35,134)( 36,136)( 37,129)( 38,131)( 39,130)( 40,132)( 41,125)( 42,127)
( 43,126)( 44,128)( 45,121)( 46,123)( 47,122)( 48,124)( 49,117)( 50,119)
( 51,118)( 52,120)( 53,113)( 54,115)( 55,114)( 56,116)( 57,109)( 58,111)
( 59,110)( 60,112)( 61,105)( 62,107)( 63,106)( 64,108)( 65,101)( 66,103)
( 67,102)( 68,104)( 69, 97)( 70, 99)( 71, 98)( 72,100)( 73, 93)( 74, 95)
( 75, 94)( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)( 81, 85)( 82, 87)
( 83, 86)( 84, 88);
s2 := Sym(166)!(  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);
s3 := Sym(166)!(165,166);
poly := sub<Sym(166)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope