Polytope of Type {3,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1088551)
Rank : 4
Schlafli Type : {3,6,4}
Number of vertices, edges, etc : 4, 48, 64, 16
Order of s₀s₁s₂s₃ : 8
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 : {3,6,4}*384b
   4-fold quotients : {3,3,4}*192
   8-fold quotients : {3,6,2}*96
   16-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s3*s2*s3> of order 2.
      8 facets:
         8 of {3,6}*48
      4 vertex figures:
         2 of 2-fold non-regular quotient of {6,4}*192b
         2 of 2-fold non-regular quotient of {6,4}*192b
   P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
      12 facets:
         8 of {3,3}*24
         4 of {3,6}*48
      4 vertex figures:
         4 of 2-fold non-regular quotient of {6,4}*192b
   P/N, where N=<s2*s3*s2*s3, s1*s2*s3*s2*s1*s3> of order 4.
      4 facets:
         4 of {3,6}*48
      4 vertex figures:
         3 of 4-fold non-regular quotient of {6,4}*192b
         1 of {6,4}*48a

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope