Polytope of Type {2,69,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,69,6}*1656
if this polytope has a name.
Group : SmallGroup(1656,129)
Rank : 4
Schlafli Type : {2,69,6}
Number of vertices, edges, etc : 2, 69, 207, 6
Order of s₀s₁s₂s₃ : 138
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) :
   3-fold quotients : {2,69,2}*552
   9-fold quotients : {2,23,2}*184
   23-fold quotients : {2,3,6}*72
   69-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 25)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)( 10, 19)( 11, 18)
( 12, 17)( 13, 16)( 14, 15)( 26, 49)( 27, 71)( 28, 70)( 29, 69)( 30, 68)
( 31, 67)( 32, 66)( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)
( 39, 59)( 40, 58)( 41, 57)( 42, 56)( 43, 55)( 44, 54)( 45, 53)( 46, 52)
( 47, 51)( 48, 50)( 72,141)( 73,163)( 74,162)( 75,161)( 76,160)( 77,159)
( 78,158)( 79,157)( 80,156)( 81,155)( 82,154)( 83,153)( 84,152)( 85,151)
( 86,150)( 87,149)( 88,148)( 89,147)( 90,146)( 91,145)( 92,144)( 93,143)
( 94,142)( 95,187)( 96,209)( 97,208)( 98,207)( 99,206)(100,205)(101,204)
(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)(109,196)
(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)(117,188)
(118,164)(119,186)(120,185)(121,184)(122,183)(123,182)(124,181)(125,180)
(126,179)(127,178)(128,177)(129,176)(130,175)(131,174)(132,173)(133,172)
(134,171)(135,170)(136,169)(137,168)(138,167)(139,166)(140,165);;
s2 := (  3, 96)(  4, 95)(  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)( 18,104)
( 19,103)( 20,102)( 21,101)( 22,100)( 23, 99)( 24, 98)( 25, 97)( 26, 73)
( 27, 72)( 28, 94)( 29, 93)( 30, 92)( 31, 91)( 32, 90)( 33, 89)( 34, 88)
( 35, 87)( 36, 86)( 37, 85)( 38, 84)( 39, 83)( 40, 82)( 41, 81)( 42, 80)
( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49,119)( 50,118)
( 51,140)( 52,139)( 53,138)( 54,137)( 55,136)( 56,135)( 57,134)( 58,133)
( 59,132)( 60,131)( 61,130)( 62,129)( 63,128)( 64,127)( 65,126)( 66,125)
( 67,124)( 68,123)( 69,122)( 70,121)( 71,120)(141,165)(142,164)(143,186)
(144,185)(145,184)(146,183)(147,182)(148,181)(149,180)(150,179)(151,178)
(152,177)(153,176)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)
(160,169)(161,168)(162,167)(163,166)(187,188)(189,209)(190,208)(191,207)
(192,206)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200);;
s3 := ( 72,141)( 73,142)( 74,143)( 75,144)( 76,145)( 77,146)( 78,147)( 79,148)
( 80,149)( 81,150)( 82,151)( 83,152)( 84,153)( 85,154)( 86,155)( 87,156)
( 88,157)( 89,158)( 90,159)( 91,160)( 92,161)( 93,162)( 94,163)( 95,164)
( 96,165)( 97,166)( 98,167)( 99,168)(100,169)(101,170)(102,171)(103,172)
(104,173)(105,174)(106,175)(107,176)(108,177)(109,178)(110,179)(111,180)
(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)
(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)
(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)
(136,205)(137,206)(138,207)(139,208)(140,209);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(209)!(1,2);
s1 := Sym(209)!(  4, 25)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)( 10, 19)
( 11, 18)( 12, 17)( 13, 16)( 14, 15)( 26, 49)( 27, 71)( 28, 70)( 29, 69)
( 30, 68)( 31, 67)( 32, 66)( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)
( 38, 60)( 39, 59)( 40, 58)( 41, 57)( 42, 56)( 43, 55)( 44, 54)( 45, 53)
( 46, 52)( 47, 51)( 48, 50)( 72,141)( 73,163)( 74,162)( 75,161)( 76,160)
( 77,159)( 78,158)( 79,157)( 80,156)( 81,155)( 82,154)( 83,153)( 84,152)
( 85,151)( 86,150)( 87,149)( 88,148)( 89,147)( 90,146)( 91,145)( 92,144)
( 93,143)( 94,142)( 95,187)( 96,209)( 97,208)( 98,207)( 99,206)(100,205)
(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)
(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)
(117,188)(118,164)(119,186)(120,185)(121,184)(122,183)(123,182)(124,181)
(125,180)(126,179)(127,178)(128,177)(129,176)(130,175)(131,174)(132,173)
(133,172)(134,171)(135,170)(136,169)(137,168)(138,167)(139,166)(140,165);
s2 := Sym(209)!(  3, 96)(  4, 95)(  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)
( 18,104)( 19,103)( 20,102)( 21,101)( 22,100)( 23, 99)( 24, 98)( 25, 97)
( 26, 73)( 27, 72)( 28, 94)( 29, 93)( 30, 92)( 31, 91)( 32, 90)( 33, 89)
( 34, 88)( 35, 87)( 36, 86)( 37, 85)( 38, 84)( 39, 83)( 40, 82)( 41, 81)
( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49,119)
( 50,118)( 51,140)( 52,139)( 53,138)( 54,137)( 55,136)( 56,135)( 57,134)
( 58,133)( 59,132)( 60,131)( 61,130)( 62,129)( 63,128)( 64,127)( 65,126)
( 66,125)( 67,124)( 68,123)( 69,122)( 70,121)( 71,120)(141,165)(142,164)
(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,180)(150,179)
(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,172)(158,171)
(159,170)(160,169)(161,168)(162,167)(163,166)(187,188)(189,209)(190,208)
(191,207)(192,206)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200);
s3 := Sym(209)!( 72,141)( 73,142)( 74,143)( 75,144)( 76,145)( 77,146)( 78,147)
( 79,148)( 80,149)( 81,150)( 82,151)( 83,152)( 84,153)( 85,154)( 86,155)
( 87,156)( 88,157)( 89,158)( 90,159)( 91,160)( 92,161)( 93,162)( 94,163)
( 95,164)( 96,165)( 97,166)( 98,167)( 99,168)(100,169)(101,170)(102,171)
(103,172)(104,173)(105,174)(106,175)(107,176)(108,177)(109,178)(110,179)
(111,180)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)
(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)
(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)
(135,204)(136,205)(137,206)(138,207)(139,208)(140,209);
poly := sub<Sym(209)|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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*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 >;

to this polytope