Polytope of Type {6,28}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,28}*1008
if this polytope has a name.
Group : SmallGroup(1008,919)
Rank : 3
Schlafli Type : {6,28}
Number of vertices, edges, etc : 18, 252, 84
Order of s0s1s2 : 28
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,28}*504
   7-fold quotients : {6,4}*144
   9-fold quotients : {2,28}*112
   14-fold quotients : {6,4}*72
   18-fold quotients : {2,14}*56
   36-fold quotients : {2,7}*28
   63-fold quotients : {2,4}*16
   126-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s1*s2> of order 3.
      56 facets:
         14 of {6}*12
         42 of {2}*4
      6 vertex figures:
         6 of {28}*56
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      28 facets:
         28 of {6}*12
      6 vertex figures:
         6 of {28}*56

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