Polytope of Type {4,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,4}*768
Also Known As : {4,3,4}(2,2,0). if this polytope has another name.
Group : SmallGroup(768,1090220)
Rank : 4
Schlafli Type : {4,3,4}
Number of vertices, edges, etc : 16, 48, 48, 16
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
Toroidal
Locally Spherical
Orientable
Self-Dual
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 : {4,3,4}*384
4-fold quotients : {4,3,4}*192a, {4,3,4}*192b
8-fold quotients : {2,3,4}*96, {4,3,2}*96, {4,3,4}*96
16-fold quotients : {2,3,4}*48, {4,3,2}*48
32-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 15)( 10, 16)( 11, 13)( 12, 14)
( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 31)( 26, 32)( 27, 29)( 28, 30)
( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)( 60, 62)
( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 79)( 74, 80)( 75, 77)( 76, 78)
( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 95)( 90, 96)( 91, 93)( 92, 94)
( 97,104)( 98,103)( 99,102)(100,101)(105,112)(106,111)(107,110)(108,109)
(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)
(129,136)(130,135)(131,134)(132,133)(137,144)(138,143)(139,142)(140,141)
(145,152)(146,151)(147,150)(148,149)(153,160)(154,159)(155,158)(156,157)
(161,168)(162,167)(163,166)(164,165)(169,176)(170,175)(171,174)(172,173)
(177,184)(178,183)(179,182)(180,181)(185,192)(186,191)(187,190)(188,189);;
s1 := ( 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);;
s2 := ( 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);;
s3 := ( 1,105)( 2,106)( 3,107)( 4,108)( 5,110)( 6,109)( 7,112)( 8,111)
( 9, 97)( 10, 98)( 11, 99)( 12,100)( 13,102)( 14,101)( 15,104)( 16,103)
( 17,121)( 18,122)( 19,123)( 20,124)( 21,126)( 22,125)( 23,128)( 24,127)
( 25,113)( 26,114)( 27,115)( 28,116)( 29,118)( 30,117)( 31,120)( 32,119)
( 33,137)( 34,138)( 35,139)( 36,140)( 37,142)( 38,141)( 39,144)( 40,143)
( 41,129)( 42,130)( 43,131)( 44,132)( 45,134)( 46,133)( 47,136)( 48,135)
( 49,153)( 50,154)( 51,155)( 52,156)( 53,158)( 54,157)( 55,160)( 56,159)
( 57,145)( 58,146)( 59,147)( 60,148)( 61,150)( 62,149)( 63,152)( 64,151)
( 65,169)( 66,170)( 67,171)( 68,172)( 69,174)( 70,173)( 71,176)( 72,175)
( 73,161)( 74,162)( 75,163)( 76,164)( 77,166)( 78,165)( 79,168)( 80,167)
( 81,185)( 82,186)( 83,187)( 84,188)( 85,190)( 86,189)( 87,192)( 88,191)
( 89,177)( 90,178)( 91,179)( 92,180)( 93,182)( 94,181)( 95,184)( 96,183);;
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, s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s3*s2*s1*s0*s1*s3*s2*s3*s1*s0*s1*s2*s3*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 15)( 10, 16)( 11, 13)
( 12, 14)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 31)( 26, 32)( 27, 29)
( 28, 30)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 47)( 42, 48)( 43, 45)
( 44, 46)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)
( 60, 62)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 79)( 74, 80)( 75, 77)
( 76, 78)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 95)( 90, 96)( 91, 93)
( 92, 94)( 97,104)( 98,103)( 99,102)(100,101)(105,112)(106,111)(107,110)
(108,109)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)
(124,125)(129,136)(130,135)(131,134)(132,133)(137,144)(138,143)(139,142)
(140,141)(145,152)(146,151)(147,150)(148,149)(153,160)(154,159)(155,158)
(156,157)(161,168)(162,167)(163,166)(164,165)(169,176)(170,175)(171,174)
(172,173)(177,184)(178,183)(179,182)(180,181)(185,192)(186,191)(187,190)
(188,189);
s1 := 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);
s2 := 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);
s3 := Sym(192)!( 1,105)( 2,106)( 3,107)( 4,108)( 5,110)( 6,109)( 7,112)
( 8,111)( 9, 97)( 10, 98)( 11, 99)( 12,100)( 13,102)( 14,101)( 15,104)
( 16,103)( 17,121)( 18,122)( 19,123)( 20,124)( 21,126)( 22,125)( 23,128)
( 24,127)( 25,113)( 26,114)( 27,115)( 28,116)( 29,118)( 30,117)( 31,120)
( 32,119)( 33,137)( 34,138)( 35,139)( 36,140)( 37,142)( 38,141)( 39,144)
( 40,143)( 41,129)( 42,130)( 43,131)( 44,132)( 45,134)( 46,133)( 47,136)
( 48,135)( 49,153)( 50,154)( 51,155)( 52,156)( 53,158)( 54,157)( 55,160)
( 56,159)( 57,145)( 58,146)( 59,147)( 60,148)( 61,150)( 62,149)( 63,152)
( 64,151)( 65,169)( 66,170)( 67,171)( 68,172)( 69,174)( 70,173)( 71,176)
( 72,175)( 73,161)( 74,162)( 75,163)( 76,164)( 77,166)( 78,165)( 79,168)
( 80,167)( 81,185)( 82,186)( 83,187)( 84,188)( 85,190)( 86,189)( 87,192)
( 88,191)( 89,177)( 90,178)( 91,179)( 92,180)( 93,182)( 94,181)( 95,184)
( 96,183);
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,
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s3*s2*s1*s0*s1*s3*s2*s3*s1*s0*s1*s2*s3*s1*s2*s0*s1 >;
References : None.
to this polytope