Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1296o
if this polytope has a name.
Group : SmallGroup(1296,2977)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 54, 324, 108
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,12}*648
   3-fold quotients : {6,4}*432a, {6,12}*432e, {6,12}*432f, {6,12}*432i
   6-fold quotients : {6,4}*216, {6,12}*216a, {6,12}*216b, {6,12}*216c
   9-fold quotients : {6,4}*144
   18-fold quotients : {6,4}*72
   27-fold quotients : {2,12}*48
   54-fold quotients : {2,6}*24
   81-fold quotients : {2,4}*16
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope