Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*864g
if this polytope has a name.
Group : SmallGroup(864,4406)
Rank : 4
Schlafli Type : {6,12,6}
Number of vertices, edges, etc : 6, 36, 36, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,12,6,2} of size 1728
Vertex Figure Of :
   {2,6,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,6}*432c
   3-fold quotients : {6,4,6}*288, {2,12,6}*288c
   4-fold quotients : {6,6,3}*216b
   6-fold quotients : {2,6,6}*144b, {6,2,6}*144
   9-fold quotients : {2,4,6}*96a, {6,4,2}*96a
   12-fold quotients : {2,6,3}*72, {3,2,6}*72, {6,2,3}*72
   18-fold quotients : {2,2,6}*48, {6,2,2}*48
   24-fold quotients : {3,2,3}*36
   27-fold quotients : {2,4,2}*32
   36-fold quotients : {2,2,3}*24, {3,2,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12,12}*1728c, {6,24,6}*1728f, {12,12,6}*1728g
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope