Polytope of Type {2,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18}*1296d
if this polytope has a name.
Group : SmallGroup(1296,1860)
Rank : 4
Schlafli Type : {2,6,18}
Number of vertices, edges, etc : 2, 18, 162, 54
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,6,9}*648b
3-fold quotients : {2,6,6}*432a
6-fold quotients : {2,6,3}*216
9-fold quotients : {2,6,6}*144b
18-fold quotients : {2,6,3}*72
27-fold quotients : {2,2,6}*48
54-fold quotients : {2,2,3}*24
81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)( 33, 63)
( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)( 41, 67)
( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)( 49, 77)
( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)(115,146)
(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)(123,153)
(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)(131,157)
(132,162)(133,164)(134,163)(135,159)(136,161)(137,160);;
s2 := ( 3, 30)( 4, 32)( 5, 31)( 6, 35)( 7, 34)( 8, 33)( 9, 37)( 10, 36)
( 11, 38)( 12, 55)( 13, 54)( 14, 56)( 15, 48)( 16, 50)( 17, 49)( 18, 53)
( 19, 52)( 20, 51)( 21, 42)( 22, 44)( 23, 43)( 24, 47)( 25, 46)( 26, 45)
( 27, 40)( 28, 39)( 29, 41)( 58, 59)( 60, 62)( 63, 64)( 66, 82)( 67, 81)
( 68, 83)( 69, 75)( 70, 77)( 71, 76)( 72, 80)( 73, 79)( 74, 78)( 84,111)
( 85,113)( 86,112)( 87,116)( 88,115)( 89,114)( 90,118)( 91,117)( 92,119)
( 93,136)( 94,135)( 95,137)( 96,129)( 97,131)( 98,130)( 99,134)(100,133)
(101,132)(102,123)(103,125)(104,124)(105,128)(106,127)(107,126)(108,121)
(109,120)(110,122)(139,140)(141,143)(144,145)(147,163)(148,162)(149,164)
(150,156)(151,158)(152,157)(153,161)(154,160)(155,159);;
s3 := ( 3, 93)( 4, 95)( 5, 94)( 6, 96)( 7, 98)( 8, 97)( 9, 99)( 10,101)
( 11,100)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)( 17, 88)( 18, 90)
( 19, 92)( 20, 91)( 21,102)( 22,104)( 23,103)( 24,105)( 25,107)( 26,106)
( 27,108)( 28,110)( 29,109)( 30,147)( 31,149)( 32,148)( 33,150)( 34,152)
( 35,151)( 36,153)( 37,155)( 38,154)( 39,138)( 40,140)( 41,139)( 42,141)
( 43,143)( 44,142)( 45,144)( 46,146)( 47,145)( 48,156)( 49,158)( 50,157)
( 51,159)( 52,161)( 53,160)( 54,162)( 55,164)( 56,163)( 57,120)( 58,122)
( 59,121)( 60,123)( 61,125)( 62,124)( 63,126)( 64,128)( 65,127)( 66,111)
( 67,113)( 68,112)( 69,114)( 70,116)( 71,115)( 72,117)( 73,119)( 74,118)
( 75,129)( 76,131)( 77,130)( 78,132)( 79,134)( 80,133)( 81,135)( 82,137)
( 83,136);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(164)!(1,2);
s1 := Sym(164)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 57)( 31, 59)( 32, 58)
( 33, 63)( 34, 65)( 35, 64)( 36, 60)( 37, 62)( 38, 61)( 39, 66)( 40, 68)
( 41, 67)( 42, 72)( 43, 74)( 44, 73)( 45, 69)( 46, 71)( 47, 70)( 48, 75)
( 49, 77)( 50, 76)( 51, 81)( 52, 83)( 53, 82)( 54, 78)( 55, 80)( 56, 79)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,138)(112,140)(113,139)(114,144)
(115,146)(116,145)(117,141)(118,143)(119,142)(120,147)(121,149)(122,148)
(123,153)(124,155)(125,154)(126,150)(127,152)(128,151)(129,156)(130,158)
(131,157)(132,162)(133,164)(134,163)(135,159)(136,161)(137,160);
s2 := Sym(164)!( 3, 30)( 4, 32)( 5, 31)( 6, 35)( 7, 34)( 8, 33)( 9, 37)
( 10, 36)( 11, 38)( 12, 55)( 13, 54)( 14, 56)( 15, 48)( 16, 50)( 17, 49)
( 18, 53)( 19, 52)( 20, 51)( 21, 42)( 22, 44)( 23, 43)( 24, 47)( 25, 46)
( 26, 45)( 27, 40)( 28, 39)( 29, 41)( 58, 59)( 60, 62)( 63, 64)( 66, 82)
( 67, 81)( 68, 83)( 69, 75)( 70, 77)( 71, 76)( 72, 80)( 73, 79)( 74, 78)
( 84,111)( 85,113)( 86,112)( 87,116)( 88,115)( 89,114)( 90,118)( 91,117)
( 92,119)( 93,136)( 94,135)( 95,137)( 96,129)( 97,131)( 98,130)( 99,134)
(100,133)(101,132)(102,123)(103,125)(104,124)(105,128)(106,127)(107,126)
(108,121)(109,120)(110,122)(139,140)(141,143)(144,145)(147,163)(148,162)
(149,164)(150,156)(151,158)(152,157)(153,161)(154,160)(155,159);
s3 := Sym(164)!( 3, 93)( 4, 95)( 5, 94)( 6, 96)( 7, 98)( 8, 97)( 9, 99)
( 10,101)( 11,100)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)( 17, 88)
( 18, 90)( 19, 92)( 20, 91)( 21,102)( 22,104)( 23,103)( 24,105)( 25,107)
( 26,106)( 27,108)( 28,110)( 29,109)( 30,147)( 31,149)( 32,148)( 33,150)
( 34,152)( 35,151)( 36,153)( 37,155)( 38,154)( 39,138)( 40,140)( 41,139)
( 42,141)( 43,143)( 44,142)( 45,144)( 46,146)( 47,145)( 48,156)( 49,158)
( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,164)( 56,163)( 57,120)
( 58,122)( 59,121)( 60,123)( 61,125)( 62,124)( 63,126)( 64,128)( 65,127)
( 66,111)( 67,113)( 68,112)( 69,114)( 70,116)( 71,115)( 72,117)( 73,119)
( 74,118)( 75,129)( 76,131)( 77,130)( 78,132)( 79,134)( 80,133)( 81,135)
( 82,137)( 83,136);
poly := sub<Sym(164)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 >;
to this polytope