Polytope of Type {2,12,9}

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

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

to this polytope