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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}*864c
if this polytope has a name.
Group : SmallGroup(864,2511)
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}*432c
   3-fold quotients : {2,6,12}*288c
   4-fold quotients : {2,3,6}*216
   6-fold quotients : {2,6,6}*144c
   9-fold quotients : {2,6,4}*96a
   12-fold quotients : {2,3,6}*72
   18-fold quotients : {2,6,2}*48
   27-fold quotients : {2,2,4}*32
   36-fold quotients : {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,12,12}*1728a, {2,6,24}*1728c, {4,6,12}*1728c
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)( 70, 82)
( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)( 93,102)
( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)(101,107);;
s2 := (  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)(  8, 76)(  9, 81)( 10, 82)
( 11, 83)( 12, 70)( 13, 71)( 14, 69)( 15, 68)( 16, 66)( 17, 67)( 18, 72)
( 19, 73)( 20, 74)( 21, 61)( 22, 62)( 23, 60)( 24, 59)( 25, 57)( 26, 58)
( 27, 63)( 28, 64)( 29, 65)( 30,106)( 31,107)( 32,105)( 33,104)( 34,102)
( 35,103)( 36,108)( 37,109)( 38,110)( 39, 97)( 40, 98)( 41, 96)( 42, 95)
( 43, 93)( 44, 94)( 45, 99)( 46,100)( 47,101)( 48, 88)( 49, 89)( 50, 87)
( 51, 86)( 52, 84)( 53, 85)( 54, 90)( 55, 91)( 56, 92);;
s3 := (  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)
( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)
( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)
( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)
( 81,105)( 82,107)( 83,106);;
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*s2*s1*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)
( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)
( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)
( 47, 53)( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)
( 70, 82)( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)
( 93,102)( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)
(101,107);
s2 := Sym(110)!(  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)(  8, 76)(  9, 81)
( 10, 82)( 11, 83)( 12, 70)( 13, 71)( 14, 69)( 15, 68)( 16, 66)( 17, 67)
( 18, 72)( 19, 73)( 20, 74)( 21, 61)( 22, 62)( 23, 60)( 24, 59)( 25, 57)
( 26, 58)( 27, 63)( 28, 64)( 29, 65)( 30,106)( 31,107)( 32,105)( 33,104)
( 34,102)( 35,103)( 36,108)( 37,109)( 38,110)( 39, 97)( 40, 98)( 41, 96)
( 42, 95)( 43, 93)( 44, 94)( 45, 99)( 46,100)( 47,101)( 48, 88)( 49, 89)
( 50, 87)( 51, 86)( 52, 84)( 53, 85)( 54, 90)( 55, 91)( 56, 92);
s3 := 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)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)
( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)
( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)
( 80,109)( 81,105)( 82,107)( 83,106);
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*s2*s1*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2 >; 
 

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