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Polytope of Type {2,18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,12}*864a
if this polytope has a name.
Group : SmallGroup(864,2438)
Rank : 4
Schlafli Type : {2,18,12}
Number of vertices, edges, etc : 2, 18, 108, 12
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,18,12,2} of size 1728
Vertex Figure Of :
   {2,2,18,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,18,6}*432a
   3-fold quotients : {2,18,4}*288a, {2,6,12}*288a
   6-fold quotients : {2,18,2}*144, {2,6,6}*144a
   9-fold quotients : {2,2,12}*96, {2,6,4}*96a
   12-fold quotients : {2,9,2}*72
   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 : {4,18,12}*1728a, {2,18,24}*1728a, {2,36,12}*1728a
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 12, 22)( 13, 21)( 14, 23)( 15, 25)( 16, 24)
( 17, 26)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 34, 35)( 37, 38)( 39, 49)
( 40, 48)( 41, 50)( 42, 52)( 43, 51)( 44, 53)( 45, 55)( 46, 54)( 47, 56)
( 58, 59)( 61, 62)( 64, 65)( 66, 76)( 67, 75)( 68, 77)( 69, 79)( 70, 78)
( 71, 80)( 72, 82)( 73, 81)( 74, 83)( 85, 86)( 88, 89)( 91, 92)( 93,103)
( 94,102)( 95,104)( 96,106)( 97,105)( 98,107)( 99,109)(100,108)(101,110);;
s2 := (  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)( 10, 17)
( 11, 16)( 21, 22)( 24, 28)( 25, 27)( 26, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 48, 49)( 51, 55)
( 52, 54)( 53, 56)( 57, 93)( 58, 95)( 59, 94)( 60, 99)( 61,101)( 62,100)
( 63, 96)( 64, 98)( 65, 97)( 66, 84)( 67, 86)( 68, 85)( 69, 90)( 70, 92)
( 71, 91)( 72, 87)( 73, 89)( 74, 88)( 75,103)( 76,102)( 77,104)( 78,109)
( 79,108)( 80,110)( 81,106)( 82,105)( 83,107);;
s3 := (  3, 60)(  4, 61)(  5, 62)(  6, 57)(  7, 58)(  8, 59)(  9, 63)( 10, 64)
( 11, 65)( 12, 69)( 13, 70)( 14, 71)( 15, 66)( 16, 67)( 17, 68)( 18, 72)
( 19, 73)( 20, 74)( 21, 78)( 22, 79)( 23, 80)( 24, 75)( 25, 76)( 26, 77)
( 27, 81)( 28, 82)( 29, 83)( 30, 87)( 31, 88)( 32, 89)( 33, 84)( 34, 85)
( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39, 96)( 40, 97)( 41, 98)( 42, 93)
( 43, 94)( 44, 95)( 45, 99)( 46,100)( 47,101)( 48,105)( 49,106)( 50,107)
( 51,102)( 52,103)( 53,104)( 54,108)( 55,109)( 56,110);;
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*s1*s2*s3*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, 
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*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  4,  5)(  7,  8)( 10, 11)( 12, 22)( 13, 21)( 14, 23)( 15, 25)
( 16, 24)( 17, 26)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 34, 35)( 37, 38)
( 39, 49)( 40, 48)( 41, 50)( 42, 52)( 43, 51)( 44, 53)( 45, 55)( 46, 54)
( 47, 56)( 58, 59)( 61, 62)( 64, 65)( 66, 76)( 67, 75)( 68, 77)( 69, 79)
( 70, 78)( 71, 80)( 72, 82)( 73, 81)( 74, 83)( 85, 86)( 88, 89)( 91, 92)
( 93,103)( 94,102)( 95,104)( 96,106)( 97,105)( 98,107)( 99,109)(100,108)
(101,110);
s2 := Sym(110)!(  3, 12)(  4, 14)(  5, 13)(  6, 18)(  7, 20)(  8, 19)(  9, 15)
( 10, 17)( 11, 16)( 21, 22)( 24, 28)( 25, 27)( 26, 29)( 30, 39)( 31, 41)
( 32, 40)( 33, 45)( 34, 47)( 35, 46)( 36, 42)( 37, 44)( 38, 43)( 48, 49)
( 51, 55)( 52, 54)( 53, 56)( 57, 93)( 58, 95)( 59, 94)( 60, 99)( 61,101)
( 62,100)( 63, 96)( 64, 98)( 65, 97)( 66, 84)( 67, 86)( 68, 85)( 69, 90)
( 70, 92)( 71, 91)( 72, 87)( 73, 89)( 74, 88)( 75,103)( 76,102)( 77,104)
( 78,109)( 79,108)( 80,110)( 81,106)( 82,105)( 83,107);
s3 := Sym(110)!(  3, 60)(  4, 61)(  5, 62)(  6, 57)(  7, 58)(  8, 59)(  9, 63)
( 10, 64)( 11, 65)( 12, 69)( 13, 70)( 14, 71)( 15, 66)( 16, 67)( 17, 68)
( 18, 72)( 19, 73)( 20, 74)( 21, 78)( 22, 79)( 23, 80)( 24, 75)( 25, 76)
( 26, 77)( 27, 81)( 28, 82)( 29, 83)( 30, 87)( 31, 88)( 32, 89)( 33, 84)
( 34, 85)( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39, 96)( 40, 97)( 41, 98)
( 42, 93)( 43, 94)( 44, 95)( 45, 99)( 46,100)( 47,101)( 48,105)( 49,106)
( 50,107)( 51,102)( 52,103)( 53,104)( 54,108)( 55,109)( 56,110);
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*s1*s2*s3*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, 
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*s1*s2*s1*s2 >; 
 

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