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Polytope of Type {2,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,12}*864b
if this polytope has a name.
Group : SmallGroup(864,2455)
Rank : 4
Schlafli Type : {2,6,12}
Number of vertices, edges, etc : 2, 18, 108, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,12,2} of size 1728
Vertex Figure Of :
   {2,2,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,6}*432b
   3-fold quotients : {2,6,12}*288a
   4-fold quotients : {2,6,6}*216
   6-fold quotients : {2,6,6}*144a
   9-fold quotients : {2,2,12}*96, {2,6,4}*96a
   18-fold quotients : {2,2,6}*48, {2,6,2}*48
   27-fold quotients : {2,2,4}*32
   36-fold quotients : {2,2,3}*24, {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,12}*1728a, {2,6,24}*1728b, {2,12,12}*1728c
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109);;
s2 := (  3,  6)(  4,  7)(  5,  8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)( 16, 22)
( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)( 39, 51)
( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)( 47, 56)
( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)
( 65, 92)( 66,105)( 67,106)( 68,107)( 69,102)( 70,103)( 71,104)( 72,108)
( 73,109)( 74,110)( 75, 96)( 76, 97)( 77, 98)( 78, 93)( 79, 94)( 80, 95)
( 81, 99)( 82,100)( 83,101);;
s3 := (  3, 66)(  4, 68)(  5, 67)(  6, 70)(  7, 69)(  8, 71)(  9, 74)( 10, 73)
( 11, 72)( 12, 57)( 13, 59)( 14, 58)( 15, 61)( 16, 60)( 17, 62)( 18, 65)
( 19, 64)( 20, 63)( 21, 75)( 22, 77)( 23, 76)( 24, 79)( 25, 78)( 26, 80)
( 27, 83)( 28, 82)( 29, 81)( 30, 93)( 31, 95)( 32, 94)( 33, 97)( 34, 96)
( 35, 98)( 36,101)( 37,100)( 38, 99)( 39, 84)( 40, 86)( 41, 85)( 42, 88)
( 43, 87)( 44, 89)( 45, 92)( 46, 91)( 47, 90)( 48,102)( 49,104)( 50,103)
( 51,106)( 52,105)( 53,107)( 54,110)( 55,109)( 56,108);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109);
s2 := Sym(110)!(  3,  6)(  4,  7)(  5,  8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)
( 16, 22)( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)
( 39, 51)( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)
( 47, 56)( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)( 63, 90)
( 64, 91)( 65, 92)( 66,105)( 67,106)( 68,107)( 69,102)( 70,103)( 71,104)
( 72,108)( 73,109)( 74,110)( 75, 96)( 76, 97)( 77, 98)( 78, 93)( 79, 94)
( 80, 95)( 81, 99)( 82,100)( 83,101);
s3 := Sym(110)!(  3, 66)(  4, 68)(  5, 67)(  6, 70)(  7, 69)(  8, 71)(  9, 74)
( 10, 73)( 11, 72)( 12, 57)( 13, 59)( 14, 58)( 15, 61)( 16, 60)( 17, 62)
( 18, 65)( 19, 64)( 20, 63)( 21, 75)( 22, 77)( 23, 76)( 24, 79)( 25, 78)
( 26, 80)( 27, 83)( 28, 82)( 29, 81)( 30, 93)( 31, 95)( 32, 94)( 33, 97)
( 34, 96)( 35, 98)( 36,101)( 37,100)( 38, 99)( 39, 84)( 40, 86)( 41, 85)
( 42, 88)( 43, 87)( 44, 89)( 45, 92)( 46, 91)( 47, 90)( 48,102)( 49,104)
( 50,103)( 51,106)( 52,105)( 53,107)( 54,110)( 55,109)( 56,108);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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