Polytope of Type {2,123,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,123,4}*1968
if this polytope has a name.
Group : SmallGroup(1968,193)
Rank : 4
Schlafli Type : {2,123,4}
Number of vertices, edges, etc : 2, 123, 246, 4
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 : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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