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Polytope of Type {2,12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,12}*1728g
if this polytope has a name.
Group : SmallGroup(1728,30413)
Rank : 4
Schlafli Type : {2,12,12}
Number of vertices, edges, etc : 2, 36, 216, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,6,12}*864e
   3-fold quotients : {2,12,4}*576
   4-fold quotients : {2,6,12}*432a
   6-fold quotients : {2,6,4}*288
   12-fold quotients : {2,6,4}*144
   27-fold quotients : {2,4,4}*64
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  3, 57)(  4, 58)(  5, 59)(  6, 65)(  7, 63)(  8, 64)(  9, 61)( 10, 62)
( 11, 60)( 12, 75)( 13, 76)( 14, 77)( 15, 83)( 16, 81)( 17, 82)( 18, 79)
( 19, 80)( 20, 78)( 21, 66)( 22, 67)( 23, 68)( 24, 74)( 25, 72)( 26, 73)
( 27, 70)( 28, 71)( 29, 69)( 30, 84)( 31, 85)( 32, 86)( 33, 92)( 34, 90)
( 35, 91)( 36, 88)( 37, 89)( 38, 87)( 39,102)( 40,103)( 41,104)( 42,110)
( 43,108)( 44,109)( 45,106)( 46,107)( 47,105)( 48, 93)( 49, 94)( 50, 95)
( 51,101)( 52, 99)( 53,100)( 54, 97)( 55, 98)( 56, 96);;
s2 := (  3, 15)(  4, 17)(  5, 16)(  6,  8)(  9, 25)( 10, 24)( 11, 26)( 13, 14)
( 18, 21)( 19, 23)( 20, 22)( 27, 29)( 30, 42)( 31, 44)( 32, 43)( 33, 35)
( 36, 52)( 37, 51)( 38, 53)( 40, 41)( 45, 48)( 46, 50)( 47, 49)( 54, 56)
( 57, 96)( 58, 98)( 59, 97)( 60, 89)( 61, 88)( 62, 87)( 63,106)( 64,105)
( 65,107)( 66, 93)( 67, 95)( 68, 94)( 69, 84)( 70, 86)( 71, 85)( 72,102)
( 73,104)( 74,103)( 75, 99)( 76,101)( 77,100)( 78, 91)( 79, 90)( 80, 92)
( 81,110)( 82,109)( 83,108);;
s3 := (  3,  5)(  7,  8)(  9, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 24)( 16, 26)
( 17, 25)( 18, 28)( 19, 27)( 20, 29)( 30, 32)( 34, 35)( 36, 37)( 39, 50)
( 40, 49)( 41, 48)( 42, 51)( 43, 53)( 44, 52)( 45, 55)( 46, 54)( 47, 56)
( 57, 59)( 61, 62)( 63, 64)( 66, 77)( 67, 76)( 68, 75)( 69, 78)( 70, 80)
( 71, 79)( 72, 82)( 73, 81)( 74, 83)( 84, 86)( 88, 89)( 90, 91)( 93,104)
( 94,103)( 95,102)( 96,105)( 97,107)( 98,106)( 99,109)(100,108)(101,110);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*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*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  3, 57)(  4, 58)(  5, 59)(  6, 65)(  7, 63)(  8, 64)(  9, 61)
( 10, 62)( 11, 60)( 12, 75)( 13, 76)( 14, 77)( 15, 83)( 16, 81)( 17, 82)
( 18, 79)( 19, 80)( 20, 78)( 21, 66)( 22, 67)( 23, 68)( 24, 74)( 25, 72)
( 26, 73)( 27, 70)( 28, 71)( 29, 69)( 30, 84)( 31, 85)( 32, 86)( 33, 92)
( 34, 90)( 35, 91)( 36, 88)( 37, 89)( 38, 87)( 39,102)( 40,103)( 41,104)
( 42,110)( 43,108)( 44,109)( 45,106)( 46,107)( 47,105)( 48, 93)( 49, 94)
( 50, 95)( 51,101)( 52, 99)( 53,100)( 54, 97)( 55, 98)( 56, 96);
s2 := Sym(110)!(  3, 15)(  4, 17)(  5, 16)(  6,  8)(  9, 25)( 10, 24)( 11, 26)
( 13, 14)( 18, 21)( 19, 23)( 20, 22)( 27, 29)( 30, 42)( 31, 44)( 32, 43)
( 33, 35)( 36, 52)( 37, 51)( 38, 53)( 40, 41)( 45, 48)( 46, 50)( 47, 49)
( 54, 56)( 57, 96)( 58, 98)( 59, 97)( 60, 89)( 61, 88)( 62, 87)( 63,106)
( 64,105)( 65,107)( 66, 93)( 67, 95)( 68, 94)( 69, 84)( 70, 86)( 71, 85)
( 72,102)( 73,104)( 74,103)( 75, 99)( 76,101)( 77,100)( 78, 91)( 79, 90)
( 80, 92)( 81,110)( 82,109)( 83,108);
s3 := Sym(110)!(  3,  5)(  7,  8)(  9, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 24)
( 16, 26)( 17, 25)( 18, 28)( 19, 27)( 20, 29)( 30, 32)( 34, 35)( 36, 37)
( 39, 50)( 40, 49)( 41, 48)( 42, 51)( 43, 53)( 44, 52)( 45, 55)( 46, 54)
( 47, 56)( 57, 59)( 61, 62)( 63, 64)( 66, 77)( 67, 76)( 68, 75)( 69, 78)
( 70, 80)( 71, 79)( 72, 82)( 73, 81)( 74, 83)( 84, 86)( 88, 89)( 90, 91)
( 93,104)( 94,103)( 95,102)( 96,105)( 97,107)( 98,106)( 99,109)(100,108)
(101,110);
poly := sub<Sym(110)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*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*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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