Polytope of Type {28,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,6}*1008
if this polytope has a name.
Group : SmallGroup(1008,919)
Rank : 3
Schlafli Type : {28,6}
Number of vertices, edges, etc : 84, 252, 18
Order of s0s1s2 : 28
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
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) :
   2-fold quotients : {28,6}*504
   7-fold quotients : {4,6}*144
   9-fold quotients : {28,2}*112
   14-fold quotients : {4,6}*72
   18-fold quotients : {14,2}*56
   36-fold quotients : {7,2}*28
   63-fold quotients : {4,2}*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=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {28}*56
      28 vertex figures:
         28 of {6}*12
   P/N, where N=<s1*s2*s1*s2> of order 3.
      6 facets:
         6 of {28}*56
      56 vertex figures:
         42 of {2}*4
         14 of {6}*12

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