Polytope of Type {6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8}*768e
if this polytope has a name.
Group : SmallGroup(768,1086320)
Rank : 3
Schlafli Type : {6,8}
Number of vertices, edges, etc : 48, 192, 64
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*384a
   4-fold quotients : {6,4}*192a, {6,8}*192b
   8-fold quotients : {3,8}*96, {6,4}*96
   16-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   32-fold quotients : {3,4}*24, {6,2}*24
   64-fold quotients : {3,2}*12
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope