Polytope of Type {6,68}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,68}*816a
Also Known As : {6,68|2}. if this polytope has another name.
Group : SmallGroup(816,133)
Rank : 3
Schlafli Type : {6,68}
Number of vertices, edges, etc : 6, 204, 68
Order of s₀s₁s₂ : 204
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,68,2} of size 1632
Vertex Figure Of :
   {2,6,68} of size 1632
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,34}*408
   3-fold quotients : {2,68}*272
   6-fold quotients : {2,34}*136
   12-fold quotients : {2,17}*68
   17-fold quotients : {6,4}*48a
   34-fold quotients : {6,2}*24
   51-fold quotients : {2,4}*16
   68-fold quotients : {3,2}*12
   102-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,136}*1632, {12,68}*1632
Permutation Representation (GAP) :

s0 := ( 18, 35)( 19, 36)( 20, 37)( 21, 38)( 22, 39)( 23, 40)( 24, 41)( 25, 42)
( 26, 43)( 27, 44)( 28, 45)( 29, 46)( 30, 47)( 31, 48)( 32, 49)( 33, 50)
( 34, 51)( 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)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)
(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)
(134,151)(135,152)(136,153)(171,188)(172,189)(173,190)(174,191)(175,192)
(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)(183,200)
(184,201)(185,202)(186,203)(187,204);;
s1 := (  1, 18)(  2, 34)(  3, 33)(  4, 32)(  5, 31)(  6, 30)(  7, 29)(  8, 28)
(  9, 27)( 10, 26)( 11, 25)( 12, 24)( 13, 23)( 14, 22)( 15, 21)( 16, 20)
( 17, 19)( 36, 51)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 43, 44)( 52, 69)( 53, 85)( 54, 84)( 55, 83)( 56, 82)( 57, 81)( 58, 80)
( 59, 79)( 60, 78)( 61, 77)( 62, 76)( 63, 75)( 64, 74)( 65, 73)( 66, 72)
( 67, 71)( 68, 70)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)
( 93, 96)( 94, 95)(103,171)(104,187)(105,186)(106,185)(107,184)(108,183)
(109,182)(110,181)(111,180)(112,179)(113,178)(114,177)(115,176)(116,175)
(117,174)(118,173)(119,172)(120,154)(121,170)(122,169)(123,168)(124,167)
(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)
(133,158)(134,157)(135,156)(136,155)(137,188)(138,204)(139,203)(140,202)
(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)
(149,193)(150,192)(151,191)(152,190)(153,189);;
s2 := (  1,104)(  2,103)(  3,119)(  4,118)(  5,117)(  6,116)(  7,115)(  8,114)
(  9,113)( 10,112)( 11,111)( 12,110)( 13,109)( 14,108)( 15,107)( 16,106)
( 17,105)( 18,121)( 19,120)( 20,136)( 21,135)( 22,134)( 23,133)( 24,132)
( 25,131)( 26,130)( 27,129)( 28,128)( 29,127)( 30,126)( 31,125)( 32,124)
( 33,123)( 34,122)( 35,138)( 36,137)( 37,153)( 38,152)( 39,151)( 40,150)
( 41,149)( 42,148)( 43,147)( 44,146)( 45,145)( 46,144)( 47,143)( 48,142)
( 49,141)( 50,140)( 51,139)( 52,155)( 53,154)( 54,170)( 55,169)( 56,168)
( 57,167)( 58,166)( 59,165)( 60,164)( 61,163)( 62,162)( 63,161)( 64,160)
( 65,159)( 66,158)( 67,157)( 68,156)( 69,172)( 70,171)( 71,187)( 72,186)
( 73,185)( 74,184)( 75,183)( 76,182)( 77,181)( 78,180)( 79,179)( 80,178)
( 81,177)( 82,176)( 83,175)( 84,174)( 85,173)( 86,189)( 87,188)( 88,204)
( 89,203)( 90,202)( 91,201)( 92,200)( 93,199)( 94,198)( 95,197)( 96,196)
( 97,195)( 98,194)( 99,193)(100,192)(101,191)(102,190);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(204)!( 18, 35)( 19, 36)( 20, 37)( 21, 38)( 22, 39)( 23, 40)( 24, 41)
( 25, 42)( 26, 43)( 27, 44)( 28, 45)( 29, 46)( 30, 47)( 31, 48)( 32, 49)
( 33, 50)( 34, 51)( 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)(120,137)(121,138)(122,139)(123,140)(124,141)
(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)
(133,150)(134,151)(135,152)(136,153)(171,188)(172,189)(173,190)(174,191)
(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)
(183,200)(184,201)(185,202)(186,203)(187,204);
s1 := Sym(204)!(  1, 18)(  2, 34)(  3, 33)(  4, 32)(  5, 31)(  6, 30)(  7, 29)
(  8, 28)(  9, 27)( 10, 26)( 11, 25)( 12, 24)( 13, 23)( 14, 22)( 15, 21)
( 16, 20)( 17, 19)( 36, 51)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 46)
( 42, 45)( 43, 44)( 52, 69)( 53, 85)( 54, 84)( 55, 83)( 56, 82)( 57, 81)
( 58, 80)( 59, 79)( 60, 78)( 61, 77)( 62, 76)( 63, 75)( 64, 74)( 65, 73)
( 66, 72)( 67, 71)( 68, 70)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)
( 92, 97)( 93, 96)( 94, 95)(103,171)(104,187)(105,186)(106,185)(107,184)
(108,183)(109,182)(110,181)(111,180)(112,179)(113,178)(114,177)(115,176)
(116,175)(117,174)(118,173)(119,172)(120,154)(121,170)(122,169)(123,168)
(124,167)(125,166)(126,165)(127,164)(128,163)(129,162)(130,161)(131,160)
(132,159)(133,158)(134,157)(135,156)(136,155)(137,188)(138,204)(139,203)
(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)
(148,194)(149,193)(150,192)(151,191)(152,190)(153,189);
s2 := Sym(204)!(  1,104)(  2,103)(  3,119)(  4,118)(  5,117)(  6,116)(  7,115)
(  8,114)(  9,113)( 10,112)( 11,111)( 12,110)( 13,109)( 14,108)( 15,107)
( 16,106)( 17,105)( 18,121)( 19,120)( 20,136)( 21,135)( 22,134)( 23,133)
( 24,132)( 25,131)( 26,130)( 27,129)( 28,128)( 29,127)( 30,126)( 31,125)
( 32,124)( 33,123)( 34,122)( 35,138)( 36,137)( 37,153)( 38,152)( 39,151)
( 40,150)( 41,149)( 42,148)( 43,147)( 44,146)( 45,145)( 46,144)( 47,143)
( 48,142)( 49,141)( 50,140)( 51,139)( 52,155)( 53,154)( 54,170)( 55,169)
( 56,168)( 57,167)( 58,166)( 59,165)( 60,164)( 61,163)( 62,162)( 63,161)
( 64,160)( 65,159)( 66,158)( 67,157)( 68,156)( 69,172)( 70,171)( 71,187)
( 72,186)( 73,185)( 74,184)( 75,183)( 76,182)( 77,181)( 78,180)( 79,179)
( 80,178)( 81,177)( 82,176)( 83,175)( 84,174)( 85,173)( 86,189)( 87,188)
( 88,204)( 89,203)( 90,202)( 91,201)( 92,200)( 93,199)( 94,198)( 95,197)
( 96,196)( 97,195)( 98,194)( 99,193)(100,192)(101,191)(102,190);
poly := sub<Sym(204)|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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 >;

References : None.
to this polytope