Polytope of Type {6,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,18}*1296c
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 4
Schlafli Type : {6,6,18}
Number of vertices, edges, etc : 6, 18, 54, 18
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 : {6,6,9}*648b
   3-fold quotients : {2,6,18}*432b, {6,2,18}*432, {6,6,6}*432c
   6-fold quotients : {2,6,9}*216, {3,2,18}*216, {6,2,9}*216, {6,6,3}*216b
   9-fold quotients : {2,2,18}*144, {2,6,6}*144b, {6,2,6}*144
   12-fold quotients : {3,2,9}*108
   18-fold quotients : {2,2,9}*72, {2,6,3}*72, {3,2,6}*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.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) :

s0 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162);;
s1 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)(  9, 33)( 10, 37)( 11, 38)( 12, 39)( 13, 43)( 14, 44)( 15, 45)( 16, 40)( 17, 41)( 18, 42)( 19, 46)( 20, 47)( 21, 48)( 22, 52)( 23, 53)( 24, 54)( 25, 49)( 26, 50)( 27, 51)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,118)( 92,119)( 93,120)( 94,124)( 95,125)( 96,126)( 97,121)( 98,122)( 99,123)(100,127)(101,128)(102,129)(103,133)(104,134)(105,135)(106,130)(107,131)(108,132)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 26)( 17, 25)( 18, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 53)( 44, 52)( 45, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 80)( 71, 79)( 72, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,104)( 92,103)( 93,105)( 94,101)( 95,100)( 96,102)( 97,107)( 98,106)( 99,108)(109,112)(110,114)(111,113)(116,117)(118,131)(119,130)(120,132)(121,128)(122,127)(123,129)(124,134)(125,133)(126,135)(136,139)(137,141)(138,140)(143,144)(145,158)(146,157)(147,159)(148,155)(149,154)(150,156)(151,161)(152,160)(153,162);;
s3 := (  1, 91)(  2, 93)(  3, 92)(  4, 97)(  5, 99)(  6, 98)(  7, 94)(  8, 96)(  9, 95)( 10, 82)( 11, 84)( 12, 83)( 13, 88)( 14, 90)( 15, 89)( 16, 85)( 17, 87)( 18, 86)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)( 24,108)( 25,104)( 26,103)( 27,105)( 28,118)( 29,120)( 30,119)( 31,124)( 32,126)( 33,125)( 34,121)( 35,123)( 36,122)( 37,109)( 38,111)( 39,110)( 40,115)( 41,117)( 42,116)( 43,112)( 44,114)( 45,113)( 46,128)( 47,127)( 48,129)( 49,134)( 50,133)( 51,135)( 52,131)( 53,130)( 54,132)( 55,145)( 56,147)( 57,146)( 58,151)( 59,153)( 60,152)( 61,148)( 62,150)( 63,149)( 64,136)( 65,138)( 66,137)( 67,142)( 68,144)( 69,143)( 70,139)( 71,141)( 72,140)( 73,155)( 74,154)( 75,156)( 76,161)( 77,160)( 78,162)( 79,158)( 80,157)( 81,159);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(162)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162);
s1 := Sym(162)!(  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)(  9, 33)( 10, 37)( 11, 38)( 12, 39)( 13, 43)( 14, 44)( 15, 45)( 16, 40)( 17, 41)( 18, 42)( 19, 46)( 20, 47)( 21, 48)( 22, 52)( 23, 53)( 24, 54)( 25, 49)( 26, 50)( 27, 51)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,118)( 92,119)( 93,120)( 94,124)( 95,125)( 96,126)( 97,121)( 98,122)( 99,123)(100,127)(101,128)(102,129)(103,133)(104,134)(105,135)(106,130)(107,131)(108,132)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162);
s2 := Sym(162)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 26)( 17, 25)( 18, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 53)( 44, 52)( 45, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 80)( 71, 79)( 72, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91,104)( 92,103)( 93,105)( 94,101)( 95,100)( 96,102)( 97,107)( 98,106)( 99,108)(109,112)(110,114)(111,113)(116,117)(118,131)(119,130)(120,132)(121,128)(122,127)(123,129)(124,134)(125,133)(126,135)(136,139)(137,141)(138,140)(143,144)(145,158)(146,157)(147,159)(148,155)(149,154)(150,156)(151,161)(152,160)(153,162);
s3 := Sym(162)!(  1, 91)(  2, 93)(  3, 92)(  4, 97)(  5, 99)(  6, 98)(  7, 94)(  8, 96)(  9, 95)( 10, 82)( 11, 84)( 12, 83)( 13, 88)( 14, 90)( 15, 89)( 16, 85)( 17, 87)( 18, 86)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)( 24,108)( 25,104)( 26,103)( 27,105)( 28,118)( 29,120)( 30,119)( 31,124)( 32,126)( 33,125)( 34,121)( 35,123)( 36,122)( 37,109)( 38,111)( 39,110)( 40,115)( 41,117)( 42,116)( 43,112)( 44,114)( 45,113)( 46,128)( 47,127)( 48,129)( 49,134)( 50,133)( 51,135)( 52,131)( 53,130)( 54,132)( 55,145)( 56,147)( 57,146)( 58,151)( 59,153)( 60,152)( 61,148)( 62,150)( 63,149)( 64,136)( 65,138)( 66,137)( 67,142)( 68,144)( 69,143)( 70,139)( 71,141)( 72,140)( 73,155)( 74,154)( 75,156)( 76,161)( 77,160)( 78,162)( 79,158)( 80,157)( 81,159);
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope