Polytope of Type {2,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30413)
Rank : 4
Schlafli Type : {2,4,12}
Number of vertices, edges, etc : 2, 36, 216, 108
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 : {2,4,12}*864a
   3-fold quotients : {2,4,4}*576
   4-fold quotients : {2,4,12}*432
   6-fold quotients : {2,4,4}*288
   12-fold quotients : {2,4,4}*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 := (  4,  5)(  6,  7)(  9, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 25)( 16, 24)
( 17, 26)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 33, 34)( 36, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 52)( 43, 51)( 44, 53)( 45, 56)( 46, 55)( 47, 54)
( 58, 59)( 60, 61)( 63, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 79)( 70, 78)
( 71, 80)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 87, 88)( 90, 92)( 93,102)
( 94,104)( 95,103)( 96,106)( 97,105)( 98,107)( 99,110)(100,109)(101,108);;
s2 := (  4,  5)(  6, 14)(  7, 13)(  8, 12)(  9, 22)( 10, 21)( 11, 23)( 15, 17)
( 18, 26)( 19, 25)( 20, 24)( 28, 29)( 31, 32)( 33, 41)( 34, 40)( 35, 39)
( 36, 49)( 37, 48)( 38, 50)( 42, 44)( 45, 53)( 46, 52)( 47, 51)( 55, 56)
( 57, 84)( 58, 86)( 59, 85)( 60, 95)( 61, 94)( 62, 93)( 63,103)( 64,102)
( 65,104)( 66, 89)( 67, 88)( 68, 87)( 69, 98)( 70, 97)( 71, 96)( 72,107)
( 73,106)( 74,105)( 75, 91)( 76, 90)( 77, 92)( 78,101)( 79,100)( 80, 99)
( 81,108)( 82,110)( 83,109);;
s3 := (  3, 64)(  4, 63)(  5, 65)(  6, 61)(  7, 60)(  8, 62)(  9, 58)( 10, 57)
( 11, 59)( 12, 73)( 13, 72)( 14, 74)( 15, 70)( 16, 69)( 17, 71)( 18, 67)
( 19, 66)( 20, 68)( 21, 82)( 22, 81)( 23, 83)( 24, 79)( 25, 78)( 26, 80)
( 27, 76)( 28, 75)( 29, 77)( 30, 91)( 31, 90)( 32, 92)( 33, 88)( 34, 87)
( 35, 89)( 36, 85)( 37, 84)( 38, 86)( 39,100)( 40, 99)( 41,101)( 42, 97)
( 43, 96)( 44, 98)( 45, 94)( 46, 93)( 47, 95)( 48,109)( 49,108)( 50,110)
( 51,106)( 52,105)( 53,107)( 54,103)( 55,102)( 56,104);;
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*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope