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Polytope of Type {12,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,2}*864c
if this polytope has a name.
Group : SmallGroup(864,2511)
Rank : 4
Schlafli Type : {12,6,2}
Number of vertices, edges, etc : 36, 108, 18, 2
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,6,2,2} of size 1728
Vertex Figure Of :
   {2,12,6,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,2}*432a
   3-fold quotients : {12,6,2}*288c
   4-fold quotients : {6,3,2}*216
   6-fold quotients : {6,6,2}*144b
   9-fold quotients : {4,6,2}*96a
   12-fold quotients : {6,3,2}*72
   18-fold quotients : {2,6,2}*48
   27-fold quotients : {4,2,2}*32
   36-fold quotients : {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12,2}*1728b, {24,6,2}*1728c, {12,6,4}*1728c
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)
( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)
( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)
( 79,103)( 80,105)( 81,104);;
s1 := (  1, 58)(  2, 59)(  3, 60)(  4, 55)(  5, 56)(  6, 57)(  7, 61)(  8, 62)
(  9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)( 16, 79)
( 17, 80)( 18, 81)( 19, 67)( 20, 68)( 21, 69)( 22, 64)( 23, 65)( 24, 66)
( 25, 70)( 26, 71)( 27, 72)( 28, 85)( 29, 86)( 30, 87)( 31, 82)( 32, 83)
( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)( 40,100)
( 41,101)( 42,102)( 43,106)( 44,107)( 45,108)( 46, 94)( 47, 95)( 48, 96)
( 49, 91)( 50, 92)( 51, 93)( 52, 97)( 53, 98)( 54, 99);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)(  8, 15)
(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)
( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)
( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107);;
s3 := (109,110);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(110)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)
( 62, 87)( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)
( 70, 94)( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)
( 78,107)( 79,103)( 80,105)( 81,104);
s1 := Sym(110)!(  1, 58)(  2, 59)(  3, 60)(  4, 55)(  5, 56)(  6, 57)(  7, 61)
(  8, 62)(  9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)
( 16, 79)( 17, 80)( 18, 81)( 19, 67)( 20, 68)( 21, 69)( 22, 64)( 23, 65)
( 24, 66)( 25, 70)( 26, 71)( 27, 72)( 28, 85)( 29, 86)( 30, 87)( 31, 82)
( 32, 83)( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)
( 40,100)( 41,101)( 42,102)( 43,106)( 44,107)( 45,108)( 46, 94)( 47, 95)
( 48, 96)( 49, 91)( 50, 92)( 51, 93)( 52, 97)( 53, 98)( 54, 99);
s2 := Sym(110)!(  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)
(  8, 15)(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)
( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)
( 51, 53)( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)
( 62, 69)( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)
( 85, 99)( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)
(105,107);
s3 := Sym(110)!(109,110);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 >; 
 

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