Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*864a
if this polytope has a name.
Group : SmallGroup(864,2282)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 36, 216, 108
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {4,12,2} of size 1728
Vertex Figure Of :
   {2,4,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*432a
   3-fold quotients : {4,4}*288
   4-fold quotients : {4,12}*216
   6-fold quotients : {4,4}*144
   12-fold quotients : {4,4}*72
   27-fold quotients : {4,4}*32
   54-fold quotients : {2,4}*16, {4,2}*16
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12}*1728b, {4,24}*1728b, {8,12}*1728b, {4,24}*1728d, {8,12}*1728c
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope