Polytope of Type {4,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,4}*384
Also Known As : {4,3,4}_(2,0,0). if this polytope has another name.
Group : SmallGroup(384,20163)
Rank : 4
Schlafli Type : {4,3,4}
Number of vertices, edges, etc : 8, 24, 24, 8
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,3,4,2} of size 768
Vertex Figure Of :
   {2,4,3,4} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,4}*192a, {4,3,4}*192b
   4-fold quotients : {2,3,4}*96, {4,3,2}*96, {4,3,4}*96
   8-fold quotients : {2,3,4}*48, {4,3,2}*48
   16-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,3,8}*768b, {8,3,4}*768b, {4,3,8}*768c, {8,3,4}*768c, {4,3,4}*768, {4,6,4}*768j, {4,6,4}*768l
   3-fold covers : {4,9,4}*1152, {4,3,12}*1152b, {12,3,4}*1152b
   5-fold covers : {4,15,4}*1920c
Permutation Representation (GAP) :

s0 := (  1,105)(  2,106)(  3,107)(  4,108)(  5,109)(  6,110)(  7,111)(  8,112)
(  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)( 16,104)
( 17,121)( 18,122)( 19,123)( 20,124)( 21,125)( 22,126)( 23,127)( 24,128)
( 25,113)( 26,114)( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)( 32,120)
( 33,137)( 34,138)( 35,139)( 36,140)( 37,141)( 38,142)( 39,143)( 40,144)
( 41,129)( 42,130)( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)( 48,136)
( 49,153)( 50,154)( 51,155)( 52,156)( 53,157)( 54,158)( 55,159)( 56,160)
( 57,145)( 58,146)( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)( 64,152)
( 65,169)( 66,170)( 67,171)( 68,172)( 69,173)( 70,174)( 71,175)( 72,176)
( 73,161)( 74,162)( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)( 80,168)
( 81,185)( 82,186)( 83,187)( 84,188)( 85,189)( 86,190)( 87,191)( 88,192)
( 89,177)( 90,178)( 91,179)( 92,180)( 93,181)( 94,182)( 95,183)( 96,184);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 33)( 18, 35)
( 19, 34)( 20, 36)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 37)( 26, 39)
( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)( 32, 48)( 50, 51)( 53, 57)
( 54, 59)( 55, 58)( 56, 60)( 62, 63)( 65, 81)( 66, 83)( 67, 82)( 68, 84)
( 69, 89)( 70, 91)( 71, 90)( 72, 92)( 73, 85)( 74, 87)( 75, 86)( 76, 88)
( 77, 93)( 78, 95)( 79, 94)( 80, 96)( 98, 99)(101,105)(102,107)(103,106)
(104,108)(110,111)(113,129)(114,131)(115,130)(116,132)(117,137)(118,139)
(119,138)(120,140)(121,133)(122,135)(123,134)(124,136)(125,141)(126,143)
(127,142)(128,144)(146,147)(149,153)(150,155)(151,154)(152,156)(158,159)
(161,177)(162,179)(163,178)(164,180)(165,185)(166,187)(167,186)(168,188)
(169,181)(170,183)(171,182)(172,184)(173,189)(174,191)(175,190)(176,192);;
s2 := (  1, 33)(  2, 34)(  3, 36)(  4, 35)(  5, 45)(  6, 46)(  7, 48)(  8, 47)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 37)( 14, 38)( 15, 40)( 16, 39)
( 19, 20)( 21, 29)( 22, 30)( 23, 32)( 24, 31)( 27, 28)( 49, 81)( 50, 82)
( 51, 84)( 52, 83)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)
( 59, 92)( 60, 91)( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 67, 68)( 69, 77)
( 70, 78)( 71, 80)( 72, 79)( 75, 76)( 97,129)( 98,130)( 99,132)(100,131)
(101,141)(102,142)(103,144)(104,143)(105,137)(106,138)(107,140)(108,139)
(109,133)(110,134)(111,136)(112,135)(115,116)(117,125)(118,126)(119,128)
(120,127)(123,124)(145,177)(146,178)(147,180)(148,179)(149,189)(150,190)
(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,181)(158,182)
(159,184)(160,183)(163,164)(165,173)(166,174)(167,176)(168,175)(171,172);;
s3 := (  1, 52)(  2, 51)(  3, 50)(  4, 49)(  5, 56)(  6, 55)(  7, 54)(  8, 53)
(  9, 60)( 10, 59)( 11, 58)( 12, 57)( 13, 64)( 14, 63)( 15, 62)( 16, 61)
( 17, 68)( 18, 67)( 19, 66)( 20, 65)( 21, 72)( 22, 71)( 23, 70)( 24, 69)
( 25, 76)( 26, 75)( 27, 74)( 28, 73)( 29, 80)( 30, 79)( 31, 78)( 32, 77)
( 33, 84)( 34, 83)( 35, 82)( 36, 81)( 37, 88)( 38, 87)( 39, 86)( 40, 85)
( 41, 92)( 42, 91)( 43, 90)( 44, 89)( 45, 96)( 46, 95)( 47, 94)( 48, 93)
( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)(104,149)
(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)(112,157)
(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)(120,165)
(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)(128,173)
(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)(136,181)
(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189);;
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, 
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(192)!(  1,105)(  2,106)(  3,107)(  4,108)(  5,109)(  6,110)(  7,111)
(  8,112)(  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)
( 16,104)( 17,121)( 18,122)( 19,123)( 20,124)( 21,125)( 22,126)( 23,127)
( 24,128)( 25,113)( 26,114)( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)
( 32,120)( 33,137)( 34,138)( 35,139)( 36,140)( 37,141)( 38,142)( 39,143)
( 40,144)( 41,129)( 42,130)( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)
( 48,136)( 49,153)( 50,154)( 51,155)( 52,156)( 53,157)( 54,158)( 55,159)
( 56,160)( 57,145)( 58,146)( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)
( 64,152)( 65,169)( 66,170)( 67,171)( 68,172)( 69,173)( 70,174)( 71,175)
( 72,176)( 73,161)( 74,162)( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)
( 80,168)( 81,185)( 82,186)( 83,187)( 84,188)( 85,189)( 86,190)( 87,191)
( 88,192)( 89,177)( 90,178)( 91,179)( 92,180)( 93,181)( 94,182)( 95,183)
( 96,184);
s1 := Sym(192)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 33)
( 18, 35)( 19, 34)( 20, 36)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 37)
( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)( 32, 48)( 50, 51)
( 53, 57)( 54, 59)( 55, 58)( 56, 60)( 62, 63)( 65, 81)( 66, 83)( 67, 82)
( 68, 84)( 69, 89)( 70, 91)( 71, 90)( 72, 92)( 73, 85)( 74, 87)( 75, 86)
( 76, 88)( 77, 93)( 78, 95)( 79, 94)( 80, 96)( 98, 99)(101,105)(102,107)
(103,106)(104,108)(110,111)(113,129)(114,131)(115,130)(116,132)(117,137)
(118,139)(119,138)(120,140)(121,133)(122,135)(123,134)(124,136)(125,141)
(126,143)(127,142)(128,144)(146,147)(149,153)(150,155)(151,154)(152,156)
(158,159)(161,177)(162,179)(163,178)(164,180)(165,185)(166,187)(167,186)
(168,188)(169,181)(170,183)(171,182)(172,184)(173,189)(174,191)(175,190)
(176,192);
s2 := Sym(192)!(  1, 33)(  2, 34)(  3, 36)(  4, 35)(  5, 45)(  6, 46)(  7, 48)
(  8, 47)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 37)( 14, 38)( 15, 40)
( 16, 39)( 19, 20)( 21, 29)( 22, 30)( 23, 32)( 24, 31)( 27, 28)( 49, 81)
( 50, 82)( 51, 84)( 52, 83)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)
( 58, 90)( 59, 92)( 60, 91)( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 67, 68)
( 69, 77)( 70, 78)( 71, 80)( 72, 79)( 75, 76)( 97,129)( 98,130)( 99,132)
(100,131)(101,141)(102,142)(103,144)(104,143)(105,137)(106,138)(107,140)
(108,139)(109,133)(110,134)(111,136)(112,135)(115,116)(117,125)(118,126)
(119,128)(120,127)(123,124)(145,177)(146,178)(147,180)(148,179)(149,189)
(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,181)
(158,182)(159,184)(160,183)(163,164)(165,173)(166,174)(167,176)(168,175)
(171,172);
s3 := Sym(192)!(  1, 52)(  2, 51)(  3, 50)(  4, 49)(  5, 56)(  6, 55)(  7, 54)
(  8, 53)(  9, 60)( 10, 59)( 11, 58)( 12, 57)( 13, 64)( 14, 63)( 15, 62)
( 16, 61)( 17, 68)( 18, 67)( 19, 66)( 20, 65)( 21, 72)( 22, 71)( 23, 70)
( 24, 69)( 25, 76)( 26, 75)( 27, 74)( 28, 73)( 29, 80)( 30, 79)( 31, 78)
( 32, 77)( 33, 84)( 34, 83)( 35, 82)( 36, 81)( 37, 88)( 38, 87)( 39, 86)
( 40, 85)( 41, 92)( 42, 91)( 43, 90)( 44, 89)( 45, 96)( 46, 95)( 47, 94)
( 48, 93)( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)
(104,149)(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)
(112,157)(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)
(120,165)(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)
(128,173)(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)
(136,181)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)
(144,189);
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, s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1 >;

References : None.
to this polytope