Polytope of Type {4,68}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,68}*544
Also Known As : {4,68|2}. if this polytope has another name.
Group : SmallGroup(544,104)
Rank : 3
Schlafli Type : {4,68}
Number of vertices, edges, etc : 4, 136, 68
Order of s₀s₁s₂ : 68
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,68,2} of size 1088
Vertex Figure Of :
   {2,4,68} of size 1088
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,68}*272, {4,34}*272
   4-fold quotients : {2,34}*136
   8-fold quotients : {2,17}*68
   17-fold quotients : {4,4}*32
   34-fold quotients : {2,4}*16, {4,2}*16
   68-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,68}*1088a, {4,136}*1088a, {8,68}*1088b, {4,136}*1088b, {4,68}*1088
   3-fold covers : {12,68}*1632, {4,204}*1632a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope