Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296m
if this polytope has a name.
Group : SmallGroup(1296,2976)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,6}*648a
   3-fold quotients : {12,6}*432i
   6-fold quotients : {12,6}*216c
   9-fold quotients : {36,2}*144, {4,6}*144
   18-fold quotients : {18,2}*72, {4,6}*72
   27-fold quotients : {12,2}*48
   36-fold quotients : {9,2}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {3,2}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)
( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 55)( 29, 57)( 30, 56)( 31, 63)
( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 64)( 38, 66)( 39, 65)
( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 73)( 47, 75)
( 48, 74)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 83, 84)
( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)
(103,108)(104,107)(105,106)(109,136)(110,138)(111,137)(112,144)(113,143)
(114,142)(115,141)(116,140)(117,139)(118,145)(119,147)(120,146)(121,153)
(122,152)(123,151)(124,150)(125,149)(126,148)(127,154)(128,156)(129,155)
(130,162)(131,161)(132,160)(133,159)(134,158)(135,157);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 58)( 11, 60)( 12, 59)( 13, 55)
( 14, 57)( 15, 56)( 16, 63)( 17, 62)( 18, 61)( 19, 31)( 20, 33)( 21, 32)
( 22, 28)( 23, 30)( 24, 29)( 25, 36)( 26, 35)( 27, 34)( 37, 76)( 38, 78)
( 39, 77)( 40, 73)( 41, 75)( 42, 74)( 43, 81)( 44, 80)( 45, 79)( 46, 49)
( 47, 51)( 48, 50)( 52, 54)( 64, 67)( 65, 69)( 66, 68)( 70, 72)( 82, 85)
( 83, 87)( 84, 86)( 88, 90)( 91,139)( 92,141)( 93,140)( 94,136)( 95,138)
( 96,137)( 97,144)( 98,143)( 99,142)(100,112)(101,114)(102,113)(103,109)
(104,111)(105,110)(106,117)(107,116)(108,115)(118,157)(119,159)(120,158)
(121,154)(122,156)(123,155)(124,162)(125,161)(126,160)(127,130)(128,132)
(129,131)(133,135)(145,148)(146,150)(147,149)(151,153);;
s2 := (  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)(  8, 98)
(  9, 99)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)
( 25,106)( 26,107)( 27,108)( 28,145)( 29,146)( 30,147)( 31,148)( 32,149)
( 33,150)( 34,151)( 35,152)( 36,153)( 37,136)( 38,137)( 39,138)( 40,139)
( 41,140)( 42,141)( 43,142)( 44,143)( 45,144)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,118)( 56,119)
( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126)( 64,109)
( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)( 70,115)( 71,116)( 72,117)
( 73,127)( 74,128)( 75,129)( 76,130)( 77,131)( 78,132)( 79,133)( 80,134)
( 81,135);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(162)!(  2,  3)(  4,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)
( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 55)( 29, 57)( 30, 56)
( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 64)( 38, 66)
( 39, 65)( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 73)
( 47, 75)( 48, 74)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)
(101,102)(103,108)(104,107)(105,106)(109,136)(110,138)(111,137)(112,144)
(113,143)(114,142)(115,141)(116,140)(117,139)(118,145)(119,147)(120,146)
(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,154)(128,156)
(129,155)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157);
s1 := Sym(162)!(  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 58)( 11, 60)( 12, 59)
( 13, 55)( 14, 57)( 15, 56)( 16, 63)( 17, 62)( 18, 61)( 19, 31)( 20, 33)
( 21, 32)( 22, 28)( 23, 30)( 24, 29)( 25, 36)( 26, 35)( 27, 34)( 37, 76)
( 38, 78)( 39, 77)( 40, 73)( 41, 75)( 42, 74)( 43, 81)( 44, 80)( 45, 79)
( 46, 49)( 47, 51)( 48, 50)( 52, 54)( 64, 67)( 65, 69)( 66, 68)( 70, 72)
( 82, 85)( 83, 87)( 84, 86)( 88, 90)( 91,139)( 92,141)( 93,140)( 94,136)
( 95,138)( 96,137)( 97,144)( 98,143)( 99,142)(100,112)(101,114)(102,113)
(103,109)(104,111)(105,110)(106,117)(107,116)(108,115)(118,157)(119,159)
(120,158)(121,154)(122,156)(123,155)(124,162)(125,161)(126,160)(127,130)
(128,132)(129,131)(133,135)(145,148)(146,150)(147,149)(151,153);
s2 := Sym(162)!(  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)
(  8, 98)(  9, 99)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)
( 24,105)( 25,106)( 26,107)( 27,108)( 28,145)( 29,146)( 30,147)( 31,148)
( 32,149)( 33,150)( 34,151)( 35,152)( 36,153)( 37,136)( 38,137)( 39,138)
( 40,139)( 41,140)( 42,141)( 43,142)( 44,143)( 45,144)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,118)
( 56,119)( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126)
( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)( 70,115)( 71,116)
( 72,117)( 73,127)( 74,128)( 75,129)( 76,130)( 77,131)( 78,132)( 79,133)
( 80,134)( 81,135);
poly := sub<Sym(162)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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*s0*s1*s0*s1 >;

References : None.
to this polytope