Polytope of Type {12,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,22}*528
Also Known As : {12,22|2}. if this polytope has another name.
Group : SmallGroup(528,107)
Rank : 3
Schlafli Type : {12,22}
Number of vertices, edges, etc : 12, 132, 22
Order of s₀s₁s₂ : 132
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,22,2} of size 1056
Vertex Figure Of :
   {2,12,22} of size 1056
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,22}*264
   3-fold quotients : {4,22}*176
   6-fold quotients : {2,22}*88
   11-fold quotients : {12,2}*48
   12-fold quotients : {2,11}*44
   22-fold quotients : {6,2}*24
   33-fold quotients : {4,2}*16
   44-fold quotients : {3,2}*12
   66-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,22}*1056, {12,44}*1056
   3-fold covers : {36,22}*1584, {12,66}*1584a, {12,66}*1584b
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
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