Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*432a
Also Known As : {36,6|2}. if this polytope has another name.
Group : SmallGroup(432,291)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 36, 108, 6
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {36,6,2} of size 864
   {36,6,3} of size 1296
   {36,6,4} of size 1728
   {36,6,3} of size 1728
   {36,6,4} of size 1728
Vertex Figure Of :
   {2,36,6} of size 864
   {4,36,6} of size 1728
   {4,36,6} of size 1728
   {4,36,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,6}*216a
   3-fold quotients : {36,2}*144, {12,6}*144a
   6-fold quotients : {18,2}*72, {6,6}*72a
   9-fold quotients : {12,2}*48, {4,6}*48a
   12-fold quotients : {9,2}*36
   18-fold quotients : {2,6}*24, {6,2}*24
   27-fold quotients : {4,2}*16
   36-fold quotients : {2,3}*12, {3,2}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {72,6}*864a, {36,12}*864a
   3-fold covers : {36,18}*1296a, {36,6}*1296b, {108,6}*1296a, {36,6}*1296l
   4-fold covers : {144,6}*1728a, {36,12}*1728a, {72,12}*1728a, {36,24}*1728c, {72,12}*1728c, {36,24}*1728d, {36,6}*1728b, {36,12}*1728e
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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