Polytope of Type {4,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1087743)
Rank : 4
Schlafli Type : {4,12,4}
Number of vertices, edges, etc : 4, 48, 48, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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 : {4,12,4}*384d
4-fold quotients : {4,12,2}*192b, {4,6,4}*192c
8-fold quotients : {4,6,2}*96c
16-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191);;
s1 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)( 51, 64)( 52, 63)
( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)( 59, 68)( 60, 67)
( 73, 85)( 74, 86)( 75, 88)( 76, 87)( 77, 93)( 78, 94)( 79, 96)( 80, 95)
( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 97,121)( 98,122)( 99,124)(100,123)
(101,129)(102,130)(103,132)(104,131)(105,125)(106,126)(107,128)(108,127)
(109,133)(110,134)(111,136)(112,135)(113,141)(114,142)(115,144)(116,143)
(117,137)(118,138)(119,140)(120,139)(145,181)(146,182)(147,184)(148,183)
(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)
(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)(163,180)(164,179)
(165,173)(166,174)(167,176)(168,175);;
s2 := ( 1,129)( 2,131)( 3,130)( 4,132)( 5,125)( 6,127)( 7,126)( 8,128)
( 9,121)( 10,123)( 11,122)( 12,124)( 13,141)( 14,143)( 15,142)( 16,144)
( 17,137)( 18,139)( 19,138)( 20,140)( 21,133)( 22,135)( 23,134)( 24,136)
( 25,105)( 26,107)( 27,106)( 28,108)( 29,101)( 30,103)( 31,102)( 32,104)
( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,117)( 38,119)( 39,118)( 40,120)
( 41,113)( 42,115)( 43,114)( 44,116)( 45,109)( 46,111)( 47,110)( 48,112)
( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)( 56,176)
( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)( 64,192)
( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)( 72,184)
( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)( 80,152)
( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)( 88,168)
( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)( 96,160);;
s3 := ( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 41)( 30, 42)( 31, 43)( 32, 44)
( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 73, 85)( 74, 86)( 75, 87)( 76, 88)
( 77, 89)( 78, 90)( 79, 91)( 80, 92)( 81, 93)( 82, 94)( 83, 95)( 84, 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)
(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)(136,172)
(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)(144,180);;
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*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191);
s1 := Sym(192)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)( 51, 64)
( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)( 59, 68)
( 60, 67)( 73, 85)( 74, 86)( 75, 88)( 76, 87)( 77, 93)( 78, 94)( 79, 96)
( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 97,121)( 98,122)( 99,124)
(100,123)(101,129)(102,130)(103,132)(104,131)(105,125)(106,126)(107,128)
(108,127)(109,133)(110,134)(111,136)(112,135)(113,141)(114,142)(115,144)
(116,143)(117,137)(118,138)(119,140)(120,139)(145,181)(146,182)(147,184)
(148,183)(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)
(156,187)(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)(163,180)
(164,179)(165,173)(166,174)(167,176)(168,175);
s2 := Sym(192)!( 1,129)( 2,131)( 3,130)( 4,132)( 5,125)( 6,127)( 7,126)
( 8,128)( 9,121)( 10,123)( 11,122)( 12,124)( 13,141)( 14,143)( 15,142)
( 16,144)( 17,137)( 18,139)( 19,138)( 20,140)( 21,133)( 22,135)( 23,134)
( 24,136)( 25,105)( 26,107)( 27,106)( 28,108)( 29,101)( 30,103)( 31,102)
( 32,104)( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,117)( 38,119)( 39,118)
( 40,120)( 41,113)( 42,115)( 43,114)( 44,116)( 45,109)( 46,111)( 47,110)
( 48,112)( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)
( 56,176)( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)
( 64,192)( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)
( 72,184)( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)
( 80,152)( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)
( 88,168)( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)
( 96,160);
s3 := Sym(192)!( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 41)( 30, 42)( 31, 43)
( 32, 44)( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 73, 85)( 74, 86)( 75, 87)
( 76, 88)( 77, 89)( 78, 90)( 79, 91)( 80, 92)( 81, 93)( 82, 94)( 83, 95)
( 84, 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)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)
(128,188)(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)
(136,172)(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)
(144,180);
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*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope