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# Polytope of Type {6,69}

Atlas Canonical Name : {6,69}*828
if this polytope has a name.
Group : SmallGroup(828,22)
Rank : 3
Schlafli Type : {6,69}
Number of vertices, edges, etc : 6, 207, 69
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 :
{6,69,2} of size 1656
Vertex Figure Of :
{2,6,69} of size 1656
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,69}*276
9-fold quotients : {2,23}*92
23-fold quotients : {6,3}*36
69-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,138}*1656c
Permutation Representation (GAP) :
```s0 := ( 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);;
s1 := (  1, 70)(  2, 92)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)(  8, 86)
(  9, 85)( 10, 84)( 11, 83)( 12, 82)( 13, 81)( 14, 80)( 15, 79)( 16, 78)
( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 72)( 23, 71)( 24,116)
( 25,138)( 26,137)( 27,136)( 28,135)( 29,134)( 30,133)( 31,132)( 32,131)
( 33,130)( 34,129)( 35,128)( 36,127)( 37,126)( 38,125)( 39,124)( 40,123)
( 41,122)( 42,121)( 43,120)( 44,119)( 45,118)( 46,117)( 47, 93)( 48,115)
( 49,114)( 50,113)( 51,112)( 52,111)( 53,110)( 54,109)( 55,108)( 56,107)
( 57,106)( 58,105)( 59,104)( 60,103)( 61,102)( 62,101)( 63,100)( 64, 99)
( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)(140,161)(141,160)(142,159)
(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)
(162,185)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)
(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)
(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186);;
s2 := (  1, 25)(  2, 24)(  3, 46)(  4, 45)(  5, 44)(  6, 43)(  7, 42)(  8, 41)
(  9, 40)( 10, 39)( 11, 38)( 12, 37)( 13, 36)( 14, 35)( 15, 34)( 16, 33)
( 17, 32)( 18, 31)( 19, 30)( 20, 29)( 21, 28)( 22, 27)( 23, 26)( 47, 48)
( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)( 55, 63)( 56, 62)
( 57, 61)( 58, 60)( 70,163)( 71,162)( 72,184)( 73,183)( 74,182)( 75,181)
( 76,180)( 77,179)( 78,178)( 79,177)( 80,176)( 81,175)( 82,174)( 83,173)
( 84,172)( 85,171)( 86,170)( 87,169)( 88,168)( 89,167)( 90,166)( 91,165)
( 92,164)( 93,140)( 94,139)( 95,161)( 96,160)( 97,159)( 98,158)( 99,157)
(100,156)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)
(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)
(116,186)(117,185)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)
(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)
(132,193)(133,192)(134,191)(135,190)(136,189)(137,188)(138,187);;
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*s0*s1*s2*s0*s1*s0*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*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope