Polytope of Type {6,4,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,4,3}*1152b
Also Known As : {{6,4|2},{4,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(1152,157559)
Rank : 5
Schlafli Type : {6,4,4,3}
Number of vertices, edges, etc : 6, 12, 16, 12, 6
Order of s₀s₁s₂s₃s₄ : 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Universal
   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 : {6,2,4,3}*576
   3-fold quotients : {2,4,4,3}*384b
   4-fold quotients : {6,4,2,3}*288a, {3,2,4,3}*288, {6,2,4,3}*288
   6-fold quotients : {2,2,4,3}*192
   8-fold quotients : {3,2,4,3}*144, {6,2,2,3}*144
   12-fold quotients : {2,4,2,3}*96, {2,2,4,3}*96
   16-fold quotients : {3,2,2,3}*72
   24-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s3*s2*s3> of order 2.
      4 facets:
         2 of {6,4,2}*96a
         2 of {6,4,4}*192
      6 vertex figures:
         6 of 2-fold non-regular quotient of {4,4,3}*192b

Permutation Representation (GAP) :

s0 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144);;
s1 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 73,113)( 74,114)( 75,115)( 76,116)( 77,109)( 78,110)( 79,111)( 80,112)( 81,117)( 82,118)( 83,119)( 84,120)( 85,125)( 86,126)( 87,127)( 88,128)( 89,121)( 90,122)( 91,123)( 92,124)( 93,129)( 94,130)( 95,131)( 96,132)( 97,137)( 98,138)( 99,139)(100,140)(101,133)(102,134)(103,135)(104,136)(105,141)(106,142)(107,143)(108,144);;
s2 := (  1, 75)(  2, 76)(  3, 73)(  4, 74)(  5, 79)(  6, 80)(  7, 77)(  8, 78)(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 87)( 14, 88)( 15, 85)( 16, 86)( 17, 91)( 18, 92)( 19, 89)( 20, 90)( 21, 95)( 22, 96)( 23, 93)( 24, 94)( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)( 32,102)( 33,107)( 34,108)( 35,105)( 36,106)( 37,111)( 38,112)( 39,109)( 40,110)( 41,115)( 42,116)( 43,113)( 44,114)( 45,119)( 46,120)( 47,117)( 48,118)( 49,123)( 50,124)( 51,121)( 52,122)( 53,127)( 54,128)( 55,125)( 56,126)( 57,131)( 58,132)( 59,129)( 60,130)( 61,135)( 62,136)( 63,133)( 64,134)( 65,139)( 66,140)( 67,137)( 68,138)( 69,143)( 70,144)( 71,141)( 72,142);;
s3 := (  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143);;
s4 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 17)(  6, 20)(  7, 19)(  8, 18)(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)(111,123)(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)(119,131)(120,130)(134,136)(138,140)(142,144);;
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(144)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144);
s1 := Sym(144)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 73,113)( 74,114)( 75,115)( 76,116)( 77,109)( 78,110)( 79,111)( 80,112)( 81,117)( 82,118)( 83,119)( 84,120)( 85,125)( 86,126)( 87,127)( 88,128)( 89,121)( 90,122)( 91,123)( 92,124)( 93,129)( 94,130)( 95,131)( 96,132)( 97,137)( 98,138)( 99,139)(100,140)(101,133)(102,134)(103,135)(104,136)(105,141)(106,142)(107,143)(108,144);
s2 := Sym(144)!(  1, 75)(  2, 76)(  3, 73)(  4, 74)(  5, 79)(  6, 80)(  7, 77)(  8, 78)(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 87)( 14, 88)( 15, 85)( 16, 86)( 17, 91)( 18, 92)( 19, 89)( 20, 90)( 21, 95)( 22, 96)( 23, 93)( 24, 94)( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)( 32,102)( 33,107)( 34,108)( 35,105)( 36,106)( 37,111)( 38,112)( 39,109)( 40,110)( 41,115)( 42,116)( 43,113)( 44,114)( 45,119)( 46,120)( 47,117)( 48,118)( 49,123)( 50,124)( 51,121)( 52,122)( 53,127)( 54,128)( 55,125)( 56,126)( 57,131)( 58,132)( 59,129)( 60,130)( 61,135)( 62,136)( 63,133)( 64,134)( 65,139)( 66,140)( 67,137)( 68,138)( 69,143)( 70,144)( 71,141)( 72,142);
s3 := Sym(144)!(  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143);
s4 := Sym(144)!(  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 17)(  6, 20)(  7, 19)(  8, 18)(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)(111,123)(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)(119,131)(120,130)(134,136)(138,140)(142,144);
poly := sub<Sym(144)|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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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