Polytope of Type {4,8,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,3}*768d
if this polytope has a name.
Group : SmallGroup(768,1088009)
Rank : 4
Schlafli Type : {4,8,3}
Number of vertices, edges, etc : 4, 64, 48, 12
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Universal
   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) :
   4-fold quotients : {4,4,3}*192b
   8-fold quotients : {2,4,3}*96
   16-fold quotients : {4,2,3}*48, {2,4,3}*48
   32-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope