Polytope of Type {4,4,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,4,3}*768a
Also Known As : {{4,4|2},{4,4}4,{4,3}3}. if this polytope has another name.
Group : SmallGroup(768,1087581)
Rank : 5
Schlafli Type : {4,4,4,3}
Number of vertices, edges, etc : 4, 16, 16, 12, 3
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {2,4,4,3}*384a
4-fold quotients : {4,2,4,3}*192
8-fold quotients : {2,2,4,3}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)
(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192);;
s1 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)( 8,103)
( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)( 16,109)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)( 24,119)
( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)( 32,125)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)( 40,135)
( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)( 56,151)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)( 64,157)
( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)( 72,167)
( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)( 80,173)
( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)( 88,183)
( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)( 96,189);;
s2 := ( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 9)( 6, 10)( 7, 11)( 8, 12)
( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)( 24, 28)
( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 37, 41)( 38, 42)( 39, 43)( 40, 44)
( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)
( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)
( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 85, 89)( 86, 90)( 87, 91)( 88, 92)
( 97,109)( 98,110)( 99,111)(100,112)(101,105)(102,106)(103,107)(104,108)
(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124)
(129,141)(130,142)(131,143)(132,144)(133,137)(134,138)(135,139)(136,140)
(145,157)(146,158)(147,159)(148,160)(149,153)(150,154)(151,155)(152,156)
(161,173)(162,174)(163,175)(164,176)(165,169)(166,170)(167,171)(168,172)
(177,189)(178,190)(179,191)(180,192)(181,185)(182,186)(183,187)(184,188);;
s3 := ( 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)( 99,100)(103,104)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)
(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)
(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)
(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s4 := ( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)( 8, 44)
( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)( 16, 48)
( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)( 50, 83)
( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)
( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 73)
( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)
(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)(108,136)
(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)(119,122)
(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)(150,187)
(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)(158,191)
(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)(174,175);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s4*s2*s3*s4*s2*s3*s4*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 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)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)
(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)
(144,192);
s1 := Sym(192)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)
( 8,103)( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)
( 16,109)( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)
( 24,119)( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)
( 32,125)( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)
( 40,135)( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)
( 56,151)( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)
( 64,157)( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)
( 72,167)( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)
( 80,173)( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)
( 88,183)( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)
( 96,189);
s2 := Sym(192)!( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 9)( 6, 10)( 7, 11)
( 8, 12)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)
( 24, 28)( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 37, 41)( 38, 42)( 39, 43)
( 40, 44)( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 57)( 54, 58)( 55, 59)
( 56, 60)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)
( 72, 76)( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 85, 89)( 86, 90)( 87, 91)
( 88, 92)( 97,109)( 98,110)( 99,111)(100,112)(101,105)(102,106)(103,107)
(104,108)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)
(120,124)(129,141)(130,142)(131,143)(132,144)(133,137)(134,138)(135,139)
(136,140)(145,157)(146,158)(147,159)(148,160)(149,153)(150,154)(151,155)
(152,156)(161,173)(162,174)(163,175)(164,176)(165,169)(166,170)(167,171)
(168,172)(177,189)(178,190)(179,191)(180,192)(181,185)(182,186)(183,187)
(184,188);
s3 := 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)( 99,100)(103,104)(105,109)
(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)
(118,134)(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)
(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)
(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)
(176,187);
s4 := Sym(192)!( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)
( 8, 44)( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)
( 16, 48)( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)
( 50, 83)( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)
( 58, 87)( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)
( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)
(100,132)(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)
(108,136)(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)
(119,122)(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)
(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)
(158,191)(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)
(174,175);
poly := sub<Sym(192)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s2*s3*s4*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
References : None.
to this polytope