Polytope of Type {33,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {33,12}*1584
if this polytope has a name.
Group : SmallGroup(1584,662)
Rank : 3
Schlafli Type : {33,12}
Number of vertices, edges, etc : 66, 396, 24
Order of s0s1s2 : 66
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {33,4}*528
   4-fold quotients : {33,6}*396
   6-fold quotients : {33,4}*264
   11-fold quotients : {3,12}*144
   12-fold quotients : {33,2}*132
   33-fold quotients : {3,4}*48
   36-fold quotients : {11,2}*44
   44-fold quotients : {3,6}*36
   66-fold quotients : {3,4}*24
   132-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      12 facets:
         12 of {33}*66
      44 vertex figures:
         22 of {12}*24
         22 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 2.
      12 facets:
         12 of {33}*66
      33 vertex figures:
         33 of {12}*24

Permutation Representation (GAP) : 
s0 := (  3,  4)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)( 24, 27)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49,129)( 50,130)( 51,132)( 52,131)( 53,125)( 54,126)( 55,128)( 56,127)( 57,121)( 58,122)( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)( 64,119)( 65,113)( 66,114)( 67,116)( 68,115)( 69,109)( 70,110)( 71,112)( 72,111)( 73,105)( 74,106)( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)( 80,103)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)( 88, 95);;
s1 := (  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 45)(  6, 48)(  7, 47)(  8, 46)(  9, 85)( 10, 88)( 11, 87)( 12, 86)( 13, 81)( 14, 84)( 15, 83)( 16, 82)( 17, 77)( 18, 80)( 19, 79)( 20, 78)( 21, 73)( 22, 76)( 23, 75)( 24, 74)( 25, 69)( 26, 72)( 27, 71)( 28, 70)( 29, 65)( 30, 68)( 31, 67)( 32, 66)( 33, 61)( 34, 64)( 35, 63)( 36, 62)( 37, 57)( 38, 60)( 39, 59)( 40, 58)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 89, 93)( 90, 96)( 91, 95)( 92, 94)( 97,129)( 98,132)( 99,131)(100,130)(101,125)(102,128)(103,127)(104,126)(105,121)(106,124)(107,123)(108,122)(109,117)(110,120)(111,119)(112,118)(114,116);;
s2 := (  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, 90)( 46, 89)( 47, 92)( 48, 91)( 49, 94)( 50, 93)( 51, 96)( 52, 95)( 53, 98)( 54, 97)( 55,100)( 56, 99)( 57,102)( 58,101)( 59,104)( 60,103)( 61,106)( 62,105)( 63,108)( 64,107)( 65,110)( 66,109)( 67,112)( 68,111)( 69,114)( 70,113)( 71,116)( 72,115)( 73,118)( 74,117)( 75,120)( 76,119)( 77,122)( 78,121)( 79,124)( 80,123)( 81,126)( 82,125)( 83,128)( 84,127)( 85,130)( 86,129)( 87,132)( 88,131);;
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s0, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(132)!(  3,  4)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)( 24, 27)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49,129)( 50,130)( 51,132)( 52,131)( 53,125)( 54,126)( 55,128)( 56,127)( 57,121)( 58,122)( 59,124)( 60,123)( 61,117)( 62,118)( 63,120)( 64,119)( 65,113)( 66,114)( 67,116)( 68,115)( 69,109)( 70,110)( 71,112)( 72,111)( 73,105)( 74,106)( 75,108)( 76,107)( 77,101)( 78,102)( 79,104)( 80,103)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)( 88, 95);
s1 := Sym(132)!(  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 45)(  6, 48)(  7, 47)(  8, 46)(  9, 85)( 10, 88)( 11, 87)( 12, 86)( 13, 81)( 14, 84)( 15, 83)( 16, 82)( 17, 77)( 18, 80)( 19, 79)( 20, 78)( 21, 73)( 22, 76)( 23, 75)( 24, 74)( 25, 69)( 26, 72)( 27, 71)( 28, 70)( 29, 65)( 30, 68)( 31, 67)( 32, 66)( 33, 61)( 34, 64)( 35, 63)( 36, 62)( 37, 57)( 38, 60)( 39, 59)( 40, 58)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 89, 93)( 90, 96)( 91, 95)( 92, 94)( 97,129)( 98,132)( 99,131)(100,130)(101,125)(102,128)(103,127)(104,126)(105,121)(106,124)(107,123)(108,122)(109,117)(110,120)(111,119)(112,118)(114,116);
s2 := Sym(132)!(  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, 90)( 46, 89)( 47, 92)( 48, 91)( 49, 94)( 50, 93)( 51, 96)( 52, 95)( 53, 98)( 54, 97)( 55,100)( 56, 99)( 57,102)( 58,101)( 59,104)( 60,103)( 61,106)( 62,105)( 63,108)( 64,107)( 65,110)( 66,109)( 67,112)( 68,111)( 69,114)( 70,113)( 71,116)( 72,115)( 73,118)( 74,117)( 75,120)( 76,119)( 77,122)( 78,121)( 79,124)( 80,123)( 81,126)( 82,125)( 83,128)( 84,127)( 85,130)( 86,129)( 87,132)( 88,131);
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s0, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*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*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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