Polytope of Type {3,8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,8,4}*768b
if this polytope has a name.
Group : SmallGroup(768,1086329)
Rank : 4
Schlafli Type : {3,8,4}
Number of vertices, edges, etc : 6, 48, 64, 8
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Orientable
   Flat
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 : {3,4,4}*384b
   4-fold quotients : {3,4,4}*192a
   8-fold quotients : {3,4,2}*96
   16-fold quotients : {3,4,2}*48
   32-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 17, 31)( 18, 32)
( 19, 29)( 20, 30)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 65)( 34, 66)
( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 76)( 42, 75)
( 43, 74)( 44, 73)( 45, 78)( 46, 77)( 47, 80)( 48, 79)( 49, 95)( 50, 96)
( 51, 93)( 52, 94)( 53, 89)( 54, 90)( 55, 91)( 56, 92)( 57, 85)( 58, 86)
( 59, 87)( 60, 88)( 61, 83)( 62, 84)( 63, 81)( 64, 82)( 97, 98)( 99,100)
(101,104)(102,103)(105,107)(106,108)(113,128)(114,127)(115,126)(116,125)
(117,122)(118,121)(119,124)(120,123)(129,162)(130,161)(131,164)(132,163)
(133,168)(134,167)(135,166)(136,165)(137,171)(138,172)(139,169)(140,170)
(141,173)(142,174)(143,175)(144,176)(145,192)(146,191)(147,190)(148,189)
(149,186)(150,185)(151,188)(152,187)(153,182)(154,181)(155,184)(156,183)
(157,180)(158,179)(159,178)(160,177);;
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 := (  1,111)(  2,112)(  3,109)(  4,110)(  5,107)(  6,108)(  7,105)(  8,106)
(  9,104)( 10,103)( 11,102)( 12,101)( 13,100)( 14, 99)( 15, 98)( 16, 97)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,123)( 22,124)( 23,121)( 24,122)
( 25,120)( 26,119)( 27,118)( 28,117)( 29,116)( 30,115)( 31,114)( 32,113)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,139)( 38,140)( 39,137)( 40,138)
( 41,136)( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)( 48,129)
( 49,159)( 50,160)( 51,157)( 52,158)( 53,155)( 54,156)( 55,153)( 56,154)
( 57,152)( 58,151)( 59,150)( 60,149)( 61,148)( 62,147)( 63,146)( 64,145)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,171)( 70,172)( 71,169)( 72,170)
( 73,168)( 74,167)( 75,166)( 76,165)( 77,164)( 78,163)( 79,162)( 80,161)
( 81,191)( 82,192)( 83,189)( 84,190)( 85,187)( 86,188)( 87,185)( 88,186)
( 89,184)( 90,183)( 91,182)( 92,181)( 93,180)( 94,179)( 95,178)( 96,177);;
s3 := (  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 63)( 58, 64)( 59, 61)( 60, 62)
( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)
( 89, 95)( 90, 96)( 91, 93)( 92, 94)( 97, 98)( 99,100)(101,102)(103,104)
(105,108)(106,107)(109,112)(110,111)(113,118)(114,117)(115,120)(116,119)
(121,128)(122,127)(123,126)(124,125)(129,130)(131,132)(133,134)(135,136)
(137,140)(138,139)(141,144)(142,143)(145,150)(146,149)(147,152)(148,151)
(153,160)(154,159)(155,158)(156,157)(161,162)(163,164)(165,166)(167,168)
(169,172)(170,171)(173,176)(174,175)(177,182)(178,181)(179,184)(180,183)
(185,192)(186,191)(187,190)(188,189);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s2*s3*s1*s2*s1*s0*s2*s1*s2*s3*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(192)!(  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 17, 31)
( 18, 32)( 19, 29)( 20, 30)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 65)
( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 76)
( 42, 75)( 43, 74)( 44, 73)( 45, 78)( 46, 77)( 47, 80)( 48, 79)( 49, 95)
( 50, 96)( 51, 93)( 52, 94)( 53, 89)( 54, 90)( 55, 91)( 56, 92)( 57, 85)
( 58, 86)( 59, 87)( 60, 88)( 61, 83)( 62, 84)( 63, 81)( 64, 82)( 97, 98)
( 99,100)(101,104)(102,103)(105,107)(106,108)(113,128)(114,127)(115,126)
(116,125)(117,122)(118,121)(119,124)(120,123)(129,162)(130,161)(131,164)
(132,163)(133,168)(134,167)(135,166)(136,165)(137,171)(138,172)(139,169)
(140,170)(141,173)(142,174)(143,175)(144,176)(145,192)(146,191)(147,190)
(148,189)(149,186)(150,185)(151,188)(152,187)(153,182)(154,181)(155,184)
(156,183)(157,180)(158,179)(159,178)(160,177);
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)!(  1,111)(  2,112)(  3,109)(  4,110)(  5,107)(  6,108)(  7,105)
(  8,106)(  9,104)( 10,103)( 11,102)( 12,101)( 13,100)( 14, 99)( 15, 98)
( 16, 97)( 17,127)( 18,128)( 19,125)( 20,126)( 21,123)( 22,124)( 23,121)
( 24,122)( 25,120)( 26,119)( 27,118)( 28,117)( 29,116)( 30,115)( 31,114)
( 32,113)( 33,143)( 34,144)( 35,141)( 36,142)( 37,139)( 38,140)( 39,137)
( 40,138)( 41,136)( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)
( 48,129)( 49,159)( 50,160)( 51,157)( 52,158)( 53,155)( 54,156)( 55,153)
( 56,154)( 57,152)( 58,151)( 59,150)( 60,149)( 61,148)( 62,147)( 63,146)
( 64,145)( 65,175)( 66,176)( 67,173)( 68,174)( 69,171)( 70,172)( 71,169)
( 72,170)( 73,168)( 74,167)( 75,166)( 76,165)( 77,164)( 78,163)( 79,162)
( 80,161)( 81,191)( 82,192)( 83,189)( 84,190)( 85,187)( 86,188)( 87,185)
( 88,186)( 89,184)( 90,183)( 91,182)( 92,181)( 93,180)( 94,179)( 95,178)
( 96,177);
s3 := Sym(192)!(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 63)( 58, 64)( 59, 61)
( 60, 62)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 85)( 82, 86)( 83, 87)
( 84, 88)( 89, 95)( 90, 96)( 91, 93)( 92, 94)( 97, 98)( 99,100)(101,102)
(103,104)(105,108)(106,107)(109,112)(110,111)(113,118)(114,117)(115,120)
(116,119)(121,128)(122,127)(123,126)(124,125)(129,130)(131,132)(133,134)
(135,136)(137,140)(138,139)(141,144)(142,143)(145,150)(146,149)(147,152)
(148,151)(153,160)(154,159)(155,158)(156,157)(161,162)(163,164)(165,166)
(167,168)(169,172)(170,171)(173,176)(174,175)(177,182)(178,181)(179,184)
(180,183)(185,192)(186,191)(187,190)(188,189);
poly := sub<Sym(192)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s2*s3*s1*s2*s1*s0*s2*s1*s2*s3*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope