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Polytope of Type {4,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,4}*192a
Also Known As : {{4,3}3,{3,4}}. if this polytope has another name.
Group : SmallGroup(192,1538)
Rank : 4
Schlafli Type : {4,3,4}
Number of vertices, edges, etc : 4, 12, 12, 8
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Locally Projective
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,3,4,2} of size 384
   {4,3,4,4} of size 768
   {4,3,4,6} of size 1152
   {4,3,4,10} of size 1920
Vertex Figure Of :
   {2,4,3,4} of size 384
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,4}*96
   4-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,3,8}*384, {4,3,4}*384, {4,6,4}*384c, {4,6,4}*384f
   3-fold covers : {4,9,4}*576a, {4,3,12}*576
   4-fold covers : {4,3,8}*768a, {4,6,4}*768g, {4,12,4}*768i, {4,12,4}*768l, {4,12,4}*768m, {4,12,4}*768n, {4,3,8}*768b, {4,6,8}*768e, {4,6,8}*768f, {8,3,4}*768b, {4,6,8}*768g, {4,3,8}*768c, {4,6,8}*768h, {8,3,4}*768c, {4,3,4}*768, {4,6,4}*768i, {4,6,4}*768j, {4,6,4}*768l
   5-fold covers : {4,15,4}*960a
   6-fold covers : {4,9,8}*1152, {4,9,4}*1152, {4,18,4}*1152c, {4,18,4}*1152f, {4,3,24}*1152, {4,3,12}*1152b, {4,6,12}*1152e, {4,6,12}*1152f, {4,6,12}*1152h, {12,3,4}*1152b, {12,6,4}*1152e
   7-fold covers : {4,21,4}*1344a
   9-fold covers : {4,27,4}*1728a, {4,9,12}*1728, {4,3,12}*1728
   10-fold covers : {4,15,8}*1920, {4,6,20}*1920c, {20,6,4}*1920c, {4,15,4}*1920c, {4,30,4}*1920c, {4,30,4}*1920f
Permutation Representation (GAP) :
s0 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192);;
s1 := (  1,113)(  2,116)(  3,115)(  4,114)(  5,121)(  6,124)(  7,123)(  8,122)
(  9,117)( 10,120)( 11,119)( 12,118)( 13,125)( 14,128)( 15,127)( 16,126)
( 17, 97)( 18,100)( 19, 99)( 20, 98)( 21,105)( 22,108)( 23,107)( 24,106)
( 25,101)( 26,104)( 27,103)( 28,102)( 29,109)( 30,112)( 31,111)( 32,110)
( 33,129)( 34,132)( 35,131)( 36,130)( 37,137)( 38,140)( 39,139)( 40,138)
( 41,133)( 42,136)( 43,135)( 44,134)( 45,141)( 46,144)( 47,143)( 48,142)
( 49,161)( 50,164)( 51,163)( 52,162)( 53,169)( 54,172)( 55,171)( 56,170)
( 57,165)( 58,168)( 59,167)( 60,166)( 61,173)( 62,176)( 63,175)( 64,174)
( 65,145)( 66,148)( 67,147)( 68,146)( 69,153)( 70,156)( 71,155)( 72,154)
( 73,149)( 74,152)( 75,151)( 76,150)( 77,157)( 78,160)( 79,159)( 80,158)
( 81,177)( 82,180)( 83,179)( 84,178)( 85,185)( 86,188)( 87,187)( 88,186)
( 89,181)( 90,184)( 91,183)( 92,182)( 93,189)( 94,192)( 95,191)( 96,190);;
s2 := (  1, 97)(  2, 98)(  3,100)(  4, 99)(  5,109)(  6,110)(  7,112)(  8,111)
(  9,105)( 10,106)( 11,108)( 12,107)( 13,101)( 14,102)( 15,104)( 16,103)
( 17,129)( 18,130)( 19,132)( 20,131)( 21,141)( 22,142)( 23,144)( 24,143)
( 25,137)( 26,138)( 27,140)( 28,139)( 29,133)( 30,134)( 31,136)( 32,135)
( 33,113)( 34,114)( 35,116)( 36,115)( 37,125)( 38,126)( 39,128)( 40,127)
( 41,121)( 42,122)( 43,124)( 44,123)( 45,117)( 46,118)( 47,120)( 48,119)
( 49,145)( 50,146)( 51,148)( 52,147)( 53,157)( 54,158)( 55,160)( 56,159)
( 57,153)( 58,154)( 59,156)( 60,155)( 61,149)( 62,150)( 63,152)( 64,151)
( 65,177)( 66,178)( 67,180)( 68,179)( 69,189)( 70,190)( 71,192)( 72,191)
( 73,185)( 74,186)( 75,188)( 76,187)( 77,181)( 78,182)( 79,184)( 80,183)
( 81,161)( 82,162)( 83,164)( 84,163)( 85,173)( 86,174)( 87,176)( 88,175)
( 89,169)( 90,170)( 91,172)( 92,171)( 93,165)( 94,166)( 95,168)( 96,167);;
s3 := (  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 57)(  6, 58)(  7, 59)(  8, 60)
(  9, 53)( 10, 54)( 11, 55)( 12, 56)( 13, 49)( 14, 50)( 15, 51)( 16, 52)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 73)( 22, 74)( 23, 75)( 24, 76)
( 25, 69)( 26, 70)( 27, 71)( 28, 72)( 29, 65)( 30, 66)( 31, 67)( 32, 68)
( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)( 40, 92)
( 41, 85)( 42, 86)( 43, 87)( 44, 88)( 45, 81)( 46, 82)( 47, 83)( 48, 84)
( 97,157)( 98,158)( 99,159)(100,160)(101,153)(102,154)(103,155)(104,156)
(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)
(113,173)(114,174)(115,175)(116,176)(117,169)(118,170)(119,171)(120,172)
(121,165)(122,166)(123,167)(124,168)(125,161)(126,162)(127,163)(128,164)
(129,189)(130,190)(131,191)(132,192)(133,185)(134,186)(135,187)(136,188)
(137,181)(138,182)(139,183)(140,184)(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, 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*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(192)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192);
s1 := Sym(192)!(  1,113)(  2,116)(  3,115)(  4,114)(  5,121)(  6,124)(  7,123)
(  8,122)(  9,117)( 10,120)( 11,119)( 12,118)( 13,125)( 14,128)( 15,127)
( 16,126)( 17, 97)( 18,100)( 19, 99)( 20, 98)( 21,105)( 22,108)( 23,107)
( 24,106)( 25,101)( 26,104)( 27,103)( 28,102)( 29,109)( 30,112)( 31,111)
( 32,110)( 33,129)( 34,132)( 35,131)( 36,130)( 37,137)( 38,140)( 39,139)
( 40,138)( 41,133)( 42,136)( 43,135)( 44,134)( 45,141)( 46,144)( 47,143)
( 48,142)( 49,161)( 50,164)( 51,163)( 52,162)( 53,169)( 54,172)( 55,171)
( 56,170)( 57,165)( 58,168)( 59,167)( 60,166)( 61,173)( 62,176)( 63,175)
( 64,174)( 65,145)( 66,148)( 67,147)( 68,146)( 69,153)( 70,156)( 71,155)
( 72,154)( 73,149)( 74,152)( 75,151)( 76,150)( 77,157)( 78,160)( 79,159)
( 80,158)( 81,177)( 82,180)( 83,179)( 84,178)( 85,185)( 86,188)( 87,187)
( 88,186)( 89,181)( 90,184)( 91,183)( 92,182)( 93,189)( 94,192)( 95,191)
( 96,190);
s2 := Sym(192)!(  1, 97)(  2, 98)(  3,100)(  4, 99)(  5,109)(  6,110)(  7,112)
(  8,111)(  9,105)( 10,106)( 11,108)( 12,107)( 13,101)( 14,102)( 15,104)
( 16,103)( 17,129)( 18,130)( 19,132)( 20,131)( 21,141)( 22,142)( 23,144)
( 24,143)( 25,137)( 26,138)( 27,140)( 28,139)( 29,133)( 30,134)( 31,136)
( 32,135)( 33,113)( 34,114)( 35,116)( 36,115)( 37,125)( 38,126)( 39,128)
( 40,127)( 41,121)( 42,122)( 43,124)( 44,123)( 45,117)( 46,118)( 47,120)
( 48,119)( 49,145)( 50,146)( 51,148)( 52,147)( 53,157)( 54,158)( 55,160)
( 56,159)( 57,153)( 58,154)( 59,156)( 60,155)( 61,149)( 62,150)( 63,152)
( 64,151)( 65,177)( 66,178)( 67,180)( 68,179)( 69,189)( 70,190)( 71,192)
( 72,191)( 73,185)( 74,186)( 75,188)( 76,187)( 77,181)( 78,182)( 79,184)
( 80,183)( 81,161)( 82,162)( 83,164)( 84,163)( 85,173)( 86,174)( 87,176)
( 88,175)( 89,169)( 90,170)( 91,172)( 92,171)( 93,165)( 94,166)( 95,168)
( 96,167);
s3 := Sym(192)!(  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 57)(  6, 58)(  7, 59)
(  8, 60)(  9, 53)( 10, 54)( 11, 55)( 12, 56)( 13, 49)( 14, 50)( 15, 51)
( 16, 52)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 73)( 22, 74)( 23, 75)
( 24, 76)( 25, 69)( 26, 70)( 27, 71)( 28, 72)( 29, 65)( 30, 66)( 31, 67)
( 32, 68)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)
( 40, 92)( 41, 85)( 42, 86)( 43, 87)( 44, 88)( 45, 81)( 46, 82)( 47, 83)
( 48, 84)( 97,157)( 98,158)( 99,159)(100,160)(101,153)(102,154)(103,155)
(104,156)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)
(112,148)(113,173)(114,174)(115,175)(116,176)(117,169)(118,170)(119,171)
(120,172)(121,165)(122,166)(123,167)(124,168)(125,161)(126,162)(127,163)
(128,164)(129,189)(130,190)(131,191)(132,192)(133,185)(134,186)(135,187)
(136,188)(137,181)(138,182)(139,183)(140,184)(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, 
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*s2*s0*s1*s2*s0*s1 >; 
 
References : None.
to this polytope