Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*864f
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}*432g
   3-fold quotients : {6,4,6}*288, {6,12,2}*288c
   4-fold quotients : {3,6,6}*216b
   6-fold quotients : {6,2,6}*144, {6,6,2}*144c
   9-fold quotients : {2,4,6}*96a, {6,4,2}*96a
   12-fold quotients : {3,2,6}*72, {3,6,2}*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 : {12,12,6}*1728f, {6,24,6}*1728g, {6,12,12}*1728g
Permutation Representation (GAP) :

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