Polytope of Type {12,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,3,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1086320)
Rank : 4
Schlafli Type : {12,3,4}
Number of vertices, edges, etc : 16, 48, 16, 8
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 : {6,3,4}*384b
4-fold quotients : {3,3,4}*192, {12,3,2}*192
8-fold quotients : {6,3,2}*96
16-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 7)( 6, 8)( 9, 10)( 11, 12)( 13, 16)( 14, 15)( 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, 74)( 42, 73)
( 43, 76)( 44, 75)( 45, 80)( 46, 79)( 47, 78)( 48, 77)( 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)( 97, 98)( 99,100)
(101,104)(102,103)(109,111)(110,112)(113,122)(114,121)(115,124)(116,123)
(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)(132,163)
(133,168)(134,167)(135,166)(136,165)(137,169)(138,170)(139,171)(140,172)
(141,175)(142,176)(143,173)(144,174)(145,186)(146,185)(147,188)(148,187)
(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)(156,179)
(157,184)(158,183)(159,182)(160,181);;
s1 := ( 1,162)( 2,161)( 3,166)( 4,165)( 5,164)( 6,163)( 7,168)( 8,167)
( 9,177)( 10,178)( 11,181)( 12,182)( 13,179)( 14,180)( 15,183)( 16,184)
( 17,169)( 18,170)( 19,173)( 20,174)( 21,171)( 22,172)( 23,175)( 24,176)
( 25,185)( 26,186)( 27,189)( 28,190)( 29,187)( 30,188)( 31,191)( 32,192)
( 33,130)( 34,129)( 35,134)( 36,133)( 37,132)( 38,131)( 39,136)( 40,135)
( 41,145)( 42,146)( 43,149)( 44,150)( 45,147)( 46,148)( 47,151)( 48,152)
( 49,137)( 50,138)( 51,141)( 52,142)( 53,139)( 54,140)( 55,143)( 56,144)
( 57,153)( 58,154)( 59,157)( 60,158)( 61,155)( 62,156)( 63,159)( 64,160)
( 65, 98)( 66, 97)( 67,102)( 68,101)( 69,100)( 70, 99)( 71,104)( 72,103)
( 73,113)( 74,114)( 75,117)( 76,118)( 77,115)( 78,116)( 79,119)( 80,120)
( 81,105)( 82,106)( 83,109)( 84,110)( 85,107)( 86,108)( 87,111)( 88,112)
( 89,121)( 90,122)( 91,125)( 92,126)( 93,123)( 94,124)( 95,127)( 96,128);;
s2 := ( 1,105)( 2,106)( 3,107)( 4,108)( 5,111)( 6,112)( 7,109)( 8,110)
( 9, 97)( 10, 98)( 11, 99)( 12,100)( 13,103)( 14,104)( 15,101)( 16,102)
( 17,114)( 18,113)( 19,116)( 20,115)( 21,120)( 22,119)( 23,118)( 24,117)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,127)( 30,128)( 31,125)( 32,126)
( 33,169)( 34,170)( 35,171)( 36,172)( 37,175)( 38,176)( 39,173)( 40,174)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,178)( 50,177)( 51,180)( 52,179)( 53,184)( 54,183)( 55,182)( 56,181)
( 57,185)( 58,186)( 59,187)( 60,188)( 61,191)( 62,192)( 63,189)( 64,190)
( 65,137)( 66,138)( 67,139)( 68,140)( 69,143)( 70,144)( 71,141)( 72,142)
( 73,129)( 74,130)( 75,131)( 76,132)( 77,135)( 78,136)( 79,133)( 80,134)
( 81,146)( 82,145)( 83,148)( 84,147)( 85,152)( 86,151)( 87,150)( 88,149)
( 89,153)( 90,154)( 91,155)( 92,156)( 93,159)( 94,160)( 95,157)( 96,158);;
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)(105,107)(106,108)(109,111)(110,112)
(113,117)(114,118)(115,119)(116,120)(121,127)(122,128)(123,125)(124,126)
(137,139)(138,140)(141,143)(142,144)(145,149)(146,150)(147,151)(148,152)
(153,159)(154,160)(155,157)(156,158)(169,171)(170,172)(173,175)(174,176)
(177,181)(178,182)(179,183)(180,184)(185,191)(186,192)(187,189)(188,190);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s0*s3*s1*s2*s3*s1*s2*s3*s1*s0*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 5, 7)( 6, 8)( 9, 10)( 11, 12)( 13, 16)( 14, 15)( 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, 74)
( 42, 73)( 43, 76)( 44, 75)( 45, 80)( 46, 79)( 47, 78)( 48, 77)( 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)( 97, 98)
( 99,100)(101,104)(102,103)(109,111)(110,112)(113,122)(114,121)(115,124)
(116,123)(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)
(132,163)(133,168)(134,167)(135,166)(136,165)(137,169)(138,170)(139,171)
(140,172)(141,175)(142,176)(143,173)(144,174)(145,186)(146,185)(147,188)
(148,187)(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)
(156,179)(157,184)(158,183)(159,182)(160,181);
s1 := Sym(192)!( 1,162)( 2,161)( 3,166)( 4,165)( 5,164)( 6,163)( 7,168)
( 8,167)( 9,177)( 10,178)( 11,181)( 12,182)( 13,179)( 14,180)( 15,183)
( 16,184)( 17,169)( 18,170)( 19,173)( 20,174)( 21,171)( 22,172)( 23,175)
( 24,176)( 25,185)( 26,186)( 27,189)( 28,190)( 29,187)( 30,188)( 31,191)
( 32,192)( 33,130)( 34,129)( 35,134)( 36,133)( 37,132)( 38,131)( 39,136)
( 40,135)( 41,145)( 42,146)( 43,149)( 44,150)( 45,147)( 46,148)( 47,151)
( 48,152)( 49,137)( 50,138)( 51,141)( 52,142)( 53,139)( 54,140)( 55,143)
( 56,144)( 57,153)( 58,154)( 59,157)( 60,158)( 61,155)( 62,156)( 63,159)
( 64,160)( 65, 98)( 66, 97)( 67,102)( 68,101)( 69,100)( 70, 99)( 71,104)
( 72,103)( 73,113)( 74,114)( 75,117)( 76,118)( 77,115)( 78,116)( 79,119)
( 80,120)( 81,105)( 82,106)( 83,109)( 84,110)( 85,107)( 86,108)( 87,111)
( 88,112)( 89,121)( 90,122)( 91,125)( 92,126)( 93,123)( 94,124)( 95,127)
( 96,128);
s2 := Sym(192)!( 1,105)( 2,106)( 3,107)( 4,108)( 5,111)( 6,112)( 7,109)
( 8,110)( 9, 97)( 10, 98)( 11, 99)( 12,100)( 13,103)( 14,104)( 15,101)
( 16,102)( 17,114)( 18,113)( 19,116)( 20,115)( 21,120)( 22,119)( 23,118)
( 24,117)( 25,121)( 26,122)( 27,123)( 28,124)( 29,127)( 30,128)( 31,125)
( 32,126)( 33,169)( 34,170)( 35,171)( 36,172)( 37,175)( 38,176)( 39,173)
( 40,174)( 41,161)( 42,162)( 43,163)( 44,164)( 45,167)( 46,168)( 47,165)
( 48,166)( 49,178)( 50,177)( 51,180)( 52,179)( 53,184)( 54,183)( 55,182)
( 56,181)( 57,185)( 58,186)( 59,187)( 60,188)( 61,191)( 62,192)( 63,189)
( 64,190)( 65,137)( 66,138)( 67,139)( 68,140)( 69,143)( 70,144)( 71,141)
( 72,142)( 73,129)( 74,130)( 75,131)( 76,132)( 77,135)( 78,136)( 79,133)
( 80,134)( 81,146)( 82,145)( 83,148)( 84,147)( 85,152)( 86,151)( 87,150)
( 88,149)( 89,153)( 90,154)( 91,155)( 92,156)( 93,159)( 94,160)( 95,157)
( 96,158);
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)(105,107)(106,108)(109,111)
(110,112)(113,117)(114,118)(115,119)(116,120)(121,127)(122,128)(123,125)
(124,126)(137,139)(138,140)(141,143)(142,144)(145,149)(146,150)(147,151)
(148,152)(153,159)(154,160)(155,157)(156,158)(169,171)(170,172)(173,175)
(174,176)(177,181)(178,182)(179,183)(180,184)(185,191)(186,192)(187,189)
(188,190);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s0*s3*s1*s2*s3*s1*s2*s3*s1*s0*s1*s2*s3*s2*s1*s2 >;
References : None.
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