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Polytope of Type {2,68,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,68,4}*1088
if this polytope has a name.
Group : SmallGroup(1088,1036)
Rank : 4
Schlafli Type : {2,68,4}
Number of vertices, edges, etc : 2, 68, 136, 4
Order of s0s1s2s3 : 68
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   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) :
   2-fold quotients : {2,68,2}*544, {2,34,4}*544
   4-fold quotients : {2,34,2}*272
   8-fold quotients : {2,17,2}*136
   17-fold quotients : {2,4,4}*64
   34-fold quotients : {2,2,4}*32, {2,4,2}*32
   68-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)( 11, 12)
( 21, 36)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)
( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 45, 46)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 71,105)( 72,121)( 73,120)( 74,119)( 75,118)( 76,117)( 77,116)( 78,115)
( 79,114)( 80,113)( 81,112)( 82,111)( 83,110)( 84,109)( 85,108)( 86,107)
( 87,106)( 88,122)( 89,138)( 90,137)( 91,136)( 92,135)( 93,134)( 94,133)
( 95,132)( 96,131)( 97,130)( 98,129)( 99,128)(100,127)(101,126)(102,125)
(103,124)(104,123);;
s2 := (  3, 72)(  4, 71)(  5, 87)(  6, 86)(  7, 85)(  8, 84)(  9, 83)( 10, 82)
( 11, 81)( 12, 80)( 13, 79)( 14, 78)( 15, 77)( 16, 76)( 17, 75)( 18, 74)
( 19, 73)( 20, 89)( 21, 88)( 22,104)( 23,103)( 24,102)( 25,101)( 26,100)
( 27, 99)( 28, 98)( 29, 97)( 30, 96)( 31, 95)( 32, 94)( 33, 93)( 34, 92)
( 35, 91)( 36, 90)( 37,106)( 38,105)( 39,121)( 40,120)( 41,119)( 42,118)
( 43,117)( 44,116)( 45,115)( 46,114)( 47,113)( 48,112)( 49,111)( 50,110)
( 51,109)( 52,108)( 53,107)( 54,123)( 55,122)( 56,138)( 57,137)( 58,136)
( 59,135)( 60,134)( 61,133)( 62,132)( 63,131)( 64,130)( 65,129)( 66,128)
( 67,127)( 68,126)( 69,125)( 70,124);;
s3 := ( 71, 88)( 72, 89)( 73, 90)( 74, 91)( 75, 92)( 76, 93)( 77, 94)( 78, 95)
( 79, 96)( 80, 97)( 81, 98)( 82, 99)( 83,100)( 84,101)( 85,102)( 86,103)
( 87,104)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)
(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)
(120,137)(121,138);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(138)!(1,2);
s1 := Sym(138)!(  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)
( 11, 12)( 21, 36)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 31)( 27, 30)
( 28, 29)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)
( 45, 46)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 71,105)( 72,121)( 73,120)( 74,119)( 75,118)( 76,117)( 77,116)
( 78,115)( 79,114)( 80,113)( 81,112)( 82,111)( 83,110)( 84,109)( 85,108)
( 86,107)( 87,106)( 88,122)( 89,138)( 90,137)( 91,136)( 92,135)( 93,134)
( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)( 99,128)(100,127)(101,126)
(102,125)(103,124)(104,123);
s2 := Sym(138)!(  3, 72)(  4, 71)(  5, 87)(  6, 86)(  7, 85)(  8, 84)(  9, 83)
( 10, 82)( 11, 81)( 12, 80)( 13, 79)( 14, 78)( 15, 77)( 16, 76)( 17, 75)
( 18, 74)( 19, 73)( 20, 89)( 21, 88)( 22,104)( 23,103)( 24,102)( 25,101)
( 26,100)( 27, 99)( 28, 98)( 29, 97)( 30, 96)( 31, 95)( 32, 94)( 33, 93)
( 34, 92)( 35, 91)( 36, 90)( 37,106)( 38,105)( 39,121)( 40,120)( 41,119)
( 42,118)( 43,117)( 44,116)( 45,115)( 46,114)( 47,113)( 48,112)( 49,111)
( 50,110)( 51,109)( 52,108)( 53,107)( 54,123)( 55,122)( 56,138)( 57,137)
( 58,136)( 59,135)( 60,134)( 61,133)( 62,132)( 63,131)( 64,130)( 65,129)
( 66,128)( 67,127)( 68,126)( 69,125)( 70,124);
s3 := Sym(138)!( 71, 88)( 72, 89)( 73, 90)( 74, 91)( 75, 92)( 76, 93)( 77, 94)
( 78, 95)( 79, 96)( 80, 97)( 81, 98)( 82, 99)( 83,100)( 84,101)( 85,102)
( 86,103)( 87,104)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)
(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)
(119,136)(120,137)(121,138);
poly := sub<Sym(138)|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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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