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# Polytope of Type {6,63}

Atlas Canonical Name : {6,63}*756
if this polytope has a name.
Group : SmallGroup(756,96)
Rank : 3
Schlafli Type : {6,63}
Number of vertices, edges, etc : 6, 189, 63
Order of s0s1s2 : 126
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,63,2} of size 1512
Vertex Figure Of :
{2,6,63} of size 1512
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,63}*252, {6,21}*252
7-fold quotients : {6,9}*108
9-fold quotients : {2,21}*84
21-fold quotients : {2,9}*36, {6,3}*36
27-fold quotients : {2,7}*28
63-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,126}*1512b
Permutation Representation (GAP) :
s0 := ( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)
( 72,135)( 73,136)( 74,137)( 75,138)( 76,139)( 77,140)( 78,141)( 79,142)
( 80,143)( 81,144)( 82,145)( 83,146)( 84,147)( 85,148)( 86,149)( 87,150)
( 88,151)( 89,152)( 90,153)( 91,154)( 92,155)( 93,156)( 94,157)( 95,158)
( 96,159)( 97,160)( 98,161)( 99,162)(100,163)(101,164)(102,165)(103,166)
(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)
(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)
(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189);;
s1 := (  1, 64)(  2, 66)(  3, 65)(  4, 82)(  5, 84)(  6, 83)(  7, 79)(  8, 81)
(  9, 80)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 70)
( 17, 72)( 18, 71)( 19, 67)( 20, 69)( 21, 68)( 22,108)( 23,107)( 24,106)
( 25,126)( 26,125)( 27,124)( 28,123)( 29,122)( 30,121)( 31,120)( 32,119)
( 33,118)( 34,117)( 35,116)( 36,115)( 37,114)( 38,113)( 39,112)( 40,111)
( 41,110)( 42,109)( 43, 87)( 44, 86)( 45, 85)( 46,105)( 47,104)( 48,103)
( 49,102)( 50,101)( 51,100)( 52, 99)( 53, 98)( 54, 97)( 55, 96)( 56, 95)
( 57, 94)( 58, 93)( 59, 92)( 60, 91)( 61, 90)( 62, 89)( 63, 88)(128,129)
(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)
(138,140)(148,171)(149,170)(150,169)(151,189)(152,188)(153,187)(154,186)
(155,185)(156,184)(157,183)(158,182)(159,181)(160,180)(161,179)(162,178)
(163,177)(164,176)(165,175)(166,174)(167,173)(168,172);;
s2 := (  1, 25)(  2, 27)(  3, 26)(  4, 22)(  5, 24)(  6, 23)(  7, 40)(  8, 42)
(  9, 41)( 10, 37)( 11, 39)( 12, 38)( 13, 34)( 14, 36)( 15, 35)( 16, 31)
( 17, 33)( 18, 32)( 19, 28)( 20, 30)( 21, 29)( 43, 48)( 44, 47)( 45, 46)
( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 64,151)
( 65,153)( 66,152)( 67,148)( 68,150)( 69,149)( 70,166)( 71,168)( 72,167)
( 73,163)( 74,165)( 75,164)( 76,160)( 77,162)( 78,161)( 79,157)( 80,159)
( 81,158)( 82,154)( 83,156)( 84,155)( 85,130)( 86,132)( 87,131)( 88,127)
( 89,129)( 90,128)( 91,145)( 92,147)( 93,146)( 94,142)( 95,144)( 96,143)
( 97,139)( 98,141)( 99,140)(100,136)(101,138)(102,137)(103,133)(104,135)
(105,134)(106,174)(107,173)(108,172)(109,171)(110,170)(111,169)(112,189)
(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)
(121,180)(122,179)(123,178)(124,177)(125,176)(126,175);;
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 ];;
poly := F / rels;;

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

References : None.
to this polytope