Polytope of Type {18,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,6}*1296c
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 : {18,6,3}*648b
3-fold quotients : {18,2,6}*432, {6,6,6}*432c
6-fold quotients : {9,2,6}*216, {18,2,3}*216, {6,6,3}*216b
9-fold quotients : {18,2,2}*144, {2,6,6}*144b, {6,2,6}*144
12-fold quotients : {9,2,3}*108
18-fold quotients : {9,2,2}*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 := ( 2, 3)( 5, 6)( 8, 9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 32, 33)( 35, 36)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 83, 84)( 86, 87)( 89, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(110,111)(113,114)(116,117)(118,128)(119,127)(120,129)(121,131)(122,130)(123,132)(124,134)(125,133)(126,135)(137,138)(140,141)(143,144)(145,155)(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)( 9, 17)( 19, 20)( 22, 23)( 25, 26)( 28, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 69)( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 46, 74)( 47, 73)( 48, 75)( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,101)(103,104)(106,107)(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)(116,153)(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)(125,144)(126,143)(127,155)(128,154)(129,156)(130,158)(131,157)(132,159)(133,161)(134,160)(135,162);;
s2 := ( 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);;
s3 := ( 1, 85)( 2, 86)( 3, 87)( 4, 82)( 5, 83)( 6, 84)( 7, 88)( 8, 89)( 9, 90)( 10, 94)( 11, 95)( 12, 96)( 13, 91)( 14, 92)( 15, 93)( 16, 97)( 17, 98)( 18, 99)( 19,103)( 20,104)( 21,105)( 22,100)( 23,101)( 24,102)( 25,106)( 26,107)( 27,108)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)( 33,138)( 34,142)( 35,143)( 36,144)( 37,148)( 38,149)( 39,150)( 40,145)( 41,146)( 42,147)( 43,151)( 44,152)( 45,153)( 46,157)( 47,158)( 48,159)( 49,154)( 50,155)( 51,156)( 52,160)( 53,161)( 54,162)( 55,112)( 56,113)( 57,114)( 58,109)( 59,110)( 60,111)( 61,115)( 62,116)( 63,117)( 64,121)( 65,122)( 66,123)( 67,118)( 68,119)( 69,120)( 70,124)( 71,125)( 72,126)( 73,130)( 74,131)( 75,132)( 76,127)( 77,128)( 78,129)( 79,133)( 80,134)( 81,135);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)( 29, 30)( 32, 33)( 35, 36)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 56, 57)( 59, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 83, 84)( 86, 87)( 89, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(110,111)(113,114)(116,117)(118,128)(119,127)(120,129)(121,131)(122,130)(123,132)(124,134)(125,133)(126,135)(137,138)(140,141)(143,144)(145,155)(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162);
s1 := Sym(162)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)( 9, 17)( 19, 20)( 22, 23)( 25, 26)( 28, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 69)( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 46, 74)( 47, 73)( 48, 75)( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,101)(103,104)(106,107)(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)(116,153)(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)(125,144)(126,143)(127,155)(128,154)(129,156)(130,158)(131,157)(132,159)(133,161)(134,160)(135,162);
s2 := 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);
s3 := Sym(162)!( 1, 85)( 2, 86)( 3, 87)( 4, 82)( 5, 83)( 6, 84)( 7, 88)( 8, 89)( 9, 90)( 10, 94)( 11, 95)( 12, 96)( 13, 91)( 14, 92)( 15, 93)( 16, 97)( 17, 98)( 18, 99)( 19,103)( 20,104)( 21,105)( 22,100)( 23,101)( 24,102)( 25,106)( 26,107)( 27,108)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)( 33,138)( 34,142)( 35,143)( 36,144)( 37,148)( 38,149)( 39,150)( 40,145)( 41,146)( 42,147)( 43,151)( 44,152)( 45,153)( 46,157)( 47,158)( 48,159)( 49,154)( 50,155)( 51,156)( 52,160)( 53,161)( 54,162)( 55,112)( 56,113)( 57,114)( 58,109)( 59,110)( 60,111)( 61,115)( 62,116)( 63,117)( 64,121)( 65,122)( 66,123)( 67,118)( 68,119)( 69,120)( 70,124)( 71,125)( 72,126)( 73,130)( 74,131)( 75,132)( 76,127)( 77,128)( 78,129)( 79,133)( 80,134)( 81,135);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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