Polytope of Type {8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6}*768g
if this polytope has a name.
Group : SmallGroup(768,1086329)
Rank : 3
Schlafli Type : {8,6}
Number of vertices, edges, etc : 64, 192, 48
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6}*384a
   4-fold quotients : {4,6}*192a, {8,6}*192c
   8-fold quotients : {4,6}*96
   16-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   32-fold quotients : {4,3}*24, {2,6}*24
   64-fold quotients : {2,3}*12
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope