Polytope of Type {69,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {69,6}*828
if this polytope has a name.
Group : SmallGroup(828,22)
Rank : 3
Schlafli Type : {69,6}
Number of vertices, edges, etc : 69, 207, 6
Order of s₀s₁s₂ : 138
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {69,6,2} of size 1656
Vertex Figure Of :
   {2,69,6} of size 1656
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {69,2}*276
   9-fold quotients : {23,2}*92
   23-fold quotients : {3,6}*36
   69-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {138,6}*1656c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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