Polytope of Type {18,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,6}*1296e
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 4
Schlafli Type : {18,6,6}
Number of vertices, edges, etc : 18, 54, 18, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,6,6}*648b
   3-fold quotients : {18,2,6}*432, {18,6,2}*432b, {6,6,6}*432g
   6-fold quotients : {9,2,6}*216, {9,6,2}*216, {18,2,3}*216, {3,6,6}*216b
   9-fold quotients : {18,2,2}*144, {6,2,6}*144, {6,6,2}*144c
   12-fold quotients : {9,2,3}*108
   18-fold quotients : {9,2,2}*72, {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
   27-fold quotients : {2,2,6}*48, {6,2,2}*48
   36-fold quotients : {3,2,3}*36
   54-fold quotients : {2,2,3}*24, {3,2,2}*24
   81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)( 28, 55)( 29, 57)( 30, 56)( 31, 58)
( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 74)( 38, 73)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 65)( 47, 64)
( 48, 66)( 49, 68)( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)( 83, 84)
( 86, 87)( 89, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)
( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,139)(113,141)
(114,140)(115,142)(116,144)(117,143)(118,155)(119,154)(120,156)(121,158)
(122,157)(123,159)(124,161)(125,160)(126,162)(127,146)(128,145)(129,147)
(130,149)(131,148)(132,150)(133,152)(134,151)(135,153);;
s1 := (  1,118)(  2,120)(  3,119)(  4,121)(  5,123)(  6,122)(  7,124)(  8,126)
(  9,125)( 10,109)( 11,111)( 12,110)( 13,112)( 14,114)( 15,113)( 16,115)
( 17,117)( 18,116)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)( 24,132)
( 25,134)( 26,133)( 27,135)( 28, 91)( 29, 93)( 30, 92)( 31, 94)( 32, 96)
( 33, 95)( 34, 97)( 35, 99)( 36, 98)( 37, 82)( 38, 84)( 39, 83)( 40, 85)
( 41, 87)( 42, 86)( 43, 88)( 44, 90)( 45, 89)( 46,101)( 47,100)( 48,102)
( 49,104)( 50,103)( 51,105)( 52,107)( 53,106)( 54,108)( 55,145)( 56,147)
( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)( 64,136)
( 65,138)( 66,137)( 67,139)( 68,141)( 69,140)( 70,142)( 71,144)( 72,143)
( 73,155)( 74,154)( 75,156)( 76,158)( 77,157)( 78,159)( 79,161)( 80,160)
( 81,162);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)
( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)
( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)
( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159);;
s3 := (  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)(109,112)(110,113)(111,114)(118,121)
(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)(138,141)
(145,148)(146,149)(147,150)(154,157)(155,158)(156,159);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)(  5,  6)(  8,  9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)
( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)
( 31, 58)( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 74)( 38, 73)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 65)
( 47, 64)( 48, 66)( 49, 68)( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)
( 83, 84)( 86, 87)( 89, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)
( 96,105)( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,139)
(113,141)(114,140)(115,142)(116,144)(117,143)(118,155)(119,154)(120,156)
(121,158)(122,157)(123,159)(124,161)(125,160)(126,162)(127,146)(128,145)
(129,147)(130,149)(131,148)(132,150)(133,152)(134,151)(135,153);
s1 := Sym(162)!(  1,118)(  2,120)(  3,119)(  4,121)(  5,123)(  6,122)(  7,124)
(  8,126)(  9,125)( 10,109)( 11,111)( 12,110)( 13,112)( 14,114)( 15,113)
( 16,115)( 17,117)( 18,116)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)
( 24,132)( 25,134)( 26,133)( 27,135)( 28, 91)( 29, 93)( 30, 92)( 31, 94)
( 32, 96)( 33, 95)( 34, 97)( 35, 99)( 36, 98)( 37, 82)( 38, 84)( 39, 83)
( 40, 85)( 41, 87)( 42, 86)( 43, 88)( 44, 90)( 45, 89)( 46,101)( 47,100)
( 48,102)( 49,104)( 50,103)( 51,105)( 52,107)( 53,106)( 54,108)( 55,145)
( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)
( 64,136)( 65,138)( 66,137)( 67,139)( 68,141)( 69,140)( 70,142)( 71,144)
( 72,143)( 73,155)( 74,154)( 75,156)( 76,158)( 77,157)( 78,159)( 79,161)
( 80,160)( 81,162);
s2 := Sym(162)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)
( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)
( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)
( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159);
s3 := Sym(162)!(  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)(109,112)(110,113)(111,114)
(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)
(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159);
poly := sub<Sym(162)|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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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