Polytope of Type {3,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,18}*648a
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 4
Schlafli Type : {3,6,18}
Number of vertices, edges, etc : 3, 9, 54, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,18,2} of size 1296
Vertex Figure Of :
   {2,3,6,18} of size 1296
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,9}*324
   3-fold quotients : {3,2,18}*216, {3,6,6}*216a
   6-fold quotients : {3,2,9}*108, {3,6,3}*108
   9-fold quotients : {3,2,6}*72
   18-fold quotients : {3,2,3}*36
   27-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,36}*1296a, {6,6,18}*1296a
   3-fold covers : {9,6,18}*1944a, {3,6,54}*1944a, {3,6,18}*1944a, {3,6,18}*1944b, {3,6,18}*1944d
Permutation Representation (GAP) :

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