Polytope of Type {2,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30413)
Rank : 4
Schlafli Type : {2,12,4}
Number of vertices, edges, etc : 2, 108, 216, 36
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,4}*864a
   3-fold quotients : {2,12,4}*576
   4-fold quotients : {2,6,4}*432
   6-fold quotients : {2,6,4}*288
   12-fold quotients : {2,6,4}*144
   27-fold quotients : {2,4,4}*64
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope