Polytope of Type {8,12,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12,3}*768
if this polytope has a name.
Group : SmallGroup(768,1086649)
Rank : 4
Schlafli Type : {8,12,3}
Number of vertices, edges, etc : 8, 64, 24, 4
Order of s₀s₁s₂s₃ : 8
Order of s₀s₁s₂s₃s₂s₁ : 2
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) :
   2-fold quotients : {4,12,3}*384, {8,6,3}*384
   4-fold quotients : {4,6,3}*192, {2,12,3}*192
   8-fold quotients : {2,6,3}*96
   16-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope