Polytope of Type {2,9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,12}*864
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {2,9,12}
Number of vertices, edges, etc : 2, 18, 108, 24
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,9,12,2} of size 1728
Vertex Figure Of :
   {2,2,9,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,9,4}*288, {2,3,12}*288
   4-fold quotients : {2,9,6}*216
   6-fold quotients : {2,9,4}*144
   9-fold quotients : {2,3,4}*96
   12-fold quotients : {2,9,2}*72, {2,3,6}*72
   18-fold quotients : {2,3,4}*48
   36-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,9,24}*1728, {2,18,12}*1728b
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 27)( 16, 29)( 17, 28)
( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)
( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)
( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)( 53,104)
( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)( 61,108)
( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)( 69, 88)
( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98);;
s2 := (  3, 51)(  4, 52)(  5, 54)(  6, 53)(  7, 59)(  8, 60)(  9, 62)( 10, 61)
( 11, 55)( 12, 56)( 13, 58)( 14, 57)( 15, 39)( 16, 40)( 17, 42)( 18, 41)
( 19, 47)( 20, 48)( 21, 50)( 22, 49)( 23, 43)( 24, 44)( 25, 46)( 26, 45)
( 27, 63)( 28, 64)( 29, 66)( 30, 65)( 31, 71)( 32, 72)( 33, 74)( 34, 73)
( 35, 67)( 36, 68)( 37, 70)( 38, 69)( 75, 91)( 76, 92)( 77, 94)( 78, 93)
( 79, 87)( 80, 88)( 81, 90)( 82, 89)( 83, 95)( 84, 96)( 85, 98)( 86, 97)
( 99,103)(100,104)(101,106)(102,105)(109,110);;
s3 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 30)( 16, 29)
( 17, 28)( 18, 27)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 38)( 24, 37)
( 25, 36)( 26, 35)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 70)( 56, 69)( 57, 68)( 58, 67)
( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91,106)( 92,105)
( 93,104)( 94,103)( 95,110)( 96,109)( 97,108)( 98,107);;
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*s1*s2*s3*s2*s1*s2, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*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, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 27)( 16, 29)
( 17, 28)( 18, 30)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)
( 25, 32)( 26, 34)( 39, 79)( 40, 81)( 41, 80)( 42, 82)( 43, 75)( 44, 77)
( 45, 76)( 46, 78)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51,103)( 52,105)
( 53,104)( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)
( 61,108)( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)
( 69, 88)( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98);
s2 := Sym(110)!(  3, 51)(  4, 52)(  5, 54)(  6, 53)(  7, 59)(  8, 60)(  9, 62)
( 10, 61)( 11, 55)( 12, 56)( 13, 58)( 14, 57)( 15, 39)( 16, 40)( 17, 42)
( 18, 41)( 19, 47)( 20, 48)( 21, 50)( 22, 49)( 23, 43)( 24, 44)( 25, 46)
( 26, 45)( 27, 63)( 28, 64)( 29, 66)( 30, 65)( 31, 71)( 32, 72)( 33, 74)
( 34, 73)( 35, 67)( 36, 68)( 37, 70)( 38, 69)( 75, 91)( 76, 92)( 77, 94)
( 78, 93)( 79, 87)( 80, 88)( 81, 90)( 82, 89)( 83, 95)( 84, 96)( 85, 98)
( 86, 97)( 99,103)(100,104)(101,106)(102,105)(109,110);
s3 := Sym(110)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 30)
( 16, 29)( 17, 28)( 18, 27)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 38)
( 24, 37)( 25, 36)( 26, 35)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 66)( 52, 65)( 53, 64)( 54, 63)( 55, 70)( 56, 69)( 57, 68)
( 58, 67)( 59, 74)( 60, 73)( 61, 72)( 62, 71)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91,106)
( 92,105)( 93,104)( 94,103)( 95,110)( 96,109)( 97,108)( 98,107);
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*s1*s2*s3*s2*s1*s2, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope