Polytope of Type {2,105,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,105,4}*1680
if this polytope has a name.
Group : SmallGroup(1680,971)
Rank : 4
Schlafli Type : {2,105,4}
Number of vertices, edges, etc : 2, 105, 210, 4
Order of s₀s₁s₂s₃ : 210
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) :
   5-fold quotients : {2,21,4}*336
   7-fold quotients : {2,15,4}*240
   35-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, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)( 13, 24)
( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31,115)( 32,117)( 33,116)
( 34,118)( 35,139)( 36,141)( 37,140)( 38,142)( 39,135)( 40,137)( 41,136)
( 42,138)( 43,131)( 44,133)( 45,132)( 46,134)( 47,127)( 48,129)( 49,128)
( 50,130)( 51,123)( 52,125)( 53,124)( 54,126)( 55,119)( 56,121)( 57,120)
( 58,122)( 59, 87)( 60, 89)( 61, 88)( 62, 90)( 63,111)( 64,113)( 65,112)
( 66,114)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)( 72,105)( 73,104)
( 74,106)( 75, 99)( 76,101)( 77,100)( 78,102)( 79, 95)( 80, 97)( 81, 96)
( 82, 98)( 83, 91)( 84, 93)( 85, 92)( 86, 94);;
s2 := (  3, 35)(  4, 38)(  5, 37)(  6, 36)(  7, 31)(  8, 34)(  9, 33)( 10, 32)
( 11, 55)( 12, 58)( 13, 57)( 14, 56)( 15, 51)( 16, 54)( 17, 53)( 18, 52)
( 19, 47)( 20, 50)( 21, 49)( 22, 48)( 23, 43)( 24, 46)( 25, 45)( 26, 44)
( 27, 39)( 28, 42)( 29, 41)( 30, 40)( 59,119)( 60,122)( 61,121)( 62,120)
( 63,115)( 64,118)( 65,117)( 66,116)( 67,139)( 68,142)( 69,141)( 70,140)
( 71,135)( 72,138)( 73,137)( 74,136)( 75,131)( 76,134)( 77,133)( 78,132)
( 79,127)( 80,130)( 81,129)( 82,128)( 83,123)( 84,126)( 85,125)( 86,124)
( 87, 91)( 88, 94)( 89, 93)( 90, 92)( 95,111)( 96,114)( 97,113)( 98,112)
( 99,107)(100,110)(101,109)(102,108)(104,106);;
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);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope