Polytope of Type {4,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,4}*768i
if this polytope has a name.
Group : SmallGroup(768,1090183)
Rank : 4
Schlafli Type : {4,12,4}
Number of vertices, edges, etc : 4, 48, 48, 8
Order of s0s1s2s3 : 6
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 : {4,6,4}*384c
4-fold quotients : {4,3,4}*192a, {4,6,4}*192d, {4,6,4}*192e
8-fold quotients : {4,6,2}*96c, {4,3,4}*96
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)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)(100,147)
(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)(108,159)
(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)(116,179)
(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)(124,191)
(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)(132,163)
(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)(140,175)
(141,169)(142,170)(143,172)(144,171);;
s2 := ( 1,129)( 2,131)( 3,130)( 4,132)( 5,137)( 6,139)( 7,138)( 8,140)
( 9,133)( 10,135)( 11,134)( 12,136)( 13,141)( 14,143)( 15,142)( 16,144)
( 17,113)( 18,115)( 19,114)( 20,116)( 21,121)( 22,123)( 23,122)( 24,124)
( 25,117)( 26,119)( 27,118)( 28,120)( 29,125)( 30,127)( 31,126)( 32,128)
( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,105)( 38,107)( 39,106)( 40,108)
( 41,101)( 42,103)( 43,102)( 44,104)( 45,109)( 46,111)( 47,110)( 48,112)
( 49,177)( 50,179)( 51,178)( 52,180)( 53,185)( 54,187)( 55,186)( 56,188)
( 57,181)( 58,183)( 59,182)( 60,184)( 61,189)( 62,191)( 63,190)( 64,192)
( 65,161)( 66,163)( 67,162)( 68,164)( 69,169)( 70,171)( 71,170)( 72,172)
( 73,165)( 74,167)( 75,166)( 76,168)( 77,173)( 78,175)( 79,174)( 80,176)
( 81,145)( 82,147)( 83,146)( 84,148)( 85,153)( 86,155)( 87,154)( 88,156)
( 89,149)( 90,151)( 91,150)( 92,152)( 93,157)( 94,159)( 95,158)( 96,160);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)(104,148)
(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)
(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)
(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)
(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)
(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188);;
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,
s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*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)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)
(100,147)(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)
(108,159)(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)
(116,179)(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)
(124,191)(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)
(132,163)(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)
(140,175)(141,169)(142,170)(143,172)(144,171);
s2 := Sym(192)!( 1,129)( 2,131)( 3,130)( 4,132)( 5,137)( 6,139)( 7,138)
( 8,140)( 9,133)( 10,135)( 11,134)( 12,136)( 13,141)( 14,143)( 15,142)
( 16,144)( 17,113)( 18,115)( 19,114)( 20,116)( 21,121)( 22,123)( 23,122)
( 24,124)( 25,117)( 26,119)( 27,118)( 28,120)( 29,125)( 30,127)( 31,126)
( 32,128)( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,105)( 38,107)( 39,106)
( 40,108)( 41,101)( 42,103)( 43,102)( 44,104)( 45,109)( 46,111)( 47,110)
( 48,112)( 49,177)( 50,179)( 51,178)( 52,180)( 53,185)( 54,187)( 55,186)
( 56,188)( 57,181)( 58,183)( 59,182)( 60,184)( 61,189)( 62,191)( 63,190)
( 64,192)( 65,161)( 66,163)( 67,162)( 68,164)( 69,169)( 70,171)( 71,170)
( 72,172)( 73,165)( 74,167)( 75,166)( 76,168)( 77,173)( 78,175)( 79,174)
( 80,176)( 81,145)( 82,147)( 83,146)( 84,148)( 85,153)( 86,155)( 87,154)
( 88,156)( 89,149)( 90,151)( 91,150)( 92,152)( 93,157)( 94,159)( 95,158)
( 96,160);
s3 := Sym(192)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)
(104,148)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)
(112,156)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)
(120,164)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)
(128,172)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)
(136,180)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)
(144,188);
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, s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;
References : None.
to this polytope