Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296o
if this polytope has a name.
Group : SmallGroup(1296,2977)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s₀s₁s₂ : 12
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 : {12,6}*648
   3-fold quotients : {4,6}*432a, {12,6}*432e, {12,6}*432f, {12,6}*432i
   6-fold quotients : {4,6}*216, {12,6}*216a, {12,6}*216b, {12,6}*216c
   9-fold quotients : {4,6}*144
   18-fold quotients : {4,6}*72
   27-fold quotients : {12,2}*48
   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,  7)(  5,  9)(  6,  8)( 10, 11)( 13, 17)( 14, 16)( 15, 18)
( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 65)( 38, 64)( 39, 66)
( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 75)( 47, 74)
( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91, 92)( 94, 98)( 95, 97)( 96, 99)(100,102)
(103,108)(104,107)(105,106)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,146)(119,145)(120,147)(121,152)
(122,151)(123,153)(124,149)(125,148)(126,150)(127,156)(128,155)(129,154)
(130,162)(131,161)(132,160)(133,159)(134,158)(135,157);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 60)( 11, 59)( 12, 58)( 13, 57)
( 14, 56)( 15, 55)( 16, 63)( 17, 62)( 18, 61)( 19, 32)( 20, 31)( 21, 33)
( 22, 29)( 23, 28)( 24, 30)( 25, 35)( 26, 34)( 27, 36)( 37, 77)( 38, 76)
( 39, 78)( 40, 74)( 41, 73)( 42, 75)( 43, 80)( 44, 79)( 45, 81)( 46, 50)
( 47, 49)( 48, 51)( 52, 53)( 64, 67)( 65, 69)( 66, 68)( 71, 72)( 82, 85)
( 83, 87)( 84, 86)( 89, 90)( 91,141)( 92,140)( 93,139)( 94,138)( 95,137)
( 96,136)( 97,144)( 98,143)( 99,142)(100,113)(101,112)(102,114)(103,110)
(104,109)(105,111)(106,116)(107,115)(108,117)(118,158)(119,157)(120,159)
(121,155)(122,154)(123,156)(124,161)(125,160)(126,162)(127,131)(128,130)
(129,132)(133,134)(145,148)(146,150)(147,149)(152,153);;
s2 := (  1,101)(  2,102)(  3,100)(  4,104)(  5,105)(  6,103)(  7,107)(  8,108)
(  9,106)( 10, 91)( 11, 92)( 12, 93)( 13, 94)( 14, 95)( 15, 96)( 16, 97)
( 17, 98)( 18, 99)( 19, 84)( 20, 82)( 21, 83)( 22, 87)( 23, 85)( 24, 86)
( 25, 90)( 26, 88)( 27, 89)( 28,155)( 29,156)( 30,154)( 31,158)( 32,159)
( 33,157)( 34,161)( 35,162)( 36,160)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,138)( 47,136)( 48,137)
( 49,141)( 50,139)( 51,140)( 52,144)( 53,142)( 54,143)( 55,128)( 56,129)
( 57,127)( 58,131)( 59,132)( 60,130)( 61,134)( 62,135)( 63,133)( 64,118)
( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)( 72,126)
( 73,111)( 74,109)( 75,110)( 76,114)( 77,112)( 78,113)( 79,117)( 80,115)
( 81,116);;
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*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope