Polytope of Type {2,18,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6,3}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 5
Schlafli Type : {2,18,6,3}
Number of vertices, edges, etc : 2, 18, 54, 9, 3
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 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 : {2,9,6,3}*648
3-fold quotients : {2,18,2,3}*432, {2,6,6,3}*432a
6-fold quotients : {2,9,2,3}*216, {2,3,6,3}*216
9-fold quotients : {2,6,2,3}*144
18-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,2,3}*48
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, 63)( 31, 65)( 32, 64)( 33, 60)
( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)
( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)
(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)
(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)
(132,159)(133,161)(134,160)(135,156)(136,158)(137,157);;
s2 := ( 3,111)( 4,113)( 5,112)( 6,117)( 7,119)( 8,118)( 9,114)( 10,116)
( 11,115)( 12,121)( 13,120)( 14,122)( 15,127)( 16,126)( 17,128)( 18,124)
( 19,123)( 20,125)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30, 84)( 31, 86)( 32, 85)( 33, 90)( 34, 92)
( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39, 94)( 40, 93)( 41, 95)( 42,100)
( 43, 99)( 44,101)( 45, 97)( 46, 96)( 47, 98)( 48,104)( 49,103)( 50,102)
( 51,110)( 52,109)( 53,108)( 54,107)( 55,106)( 56,105)( 57,144)( 58,146)
( 59,145)( 60,141)( 61,143)( 62,142)( 63,138)( 64,140)( 65,139)( 66,154)
( 67,153)( 68,155)( 69,151)( 70,150)( 71,152)( 72,148)( 73,147)( 74,149)
( 75,164)( 76,163)( 77,162)( 78,161)( 79,160)( 80,159)( 81,158)( 82,157)
( 83,156);;
s3 := ( 3, 12)( 4, 14)( 5, 13)( 6, 15)( 7, 17)( 8, 16)( 9, 18)( 10, 20)
( 11, 19)( 22, 23)( 25, 26)( 28, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 42)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 49, 50)( 52, 53)( 55, 56)
( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)( 64, 74)
( 65, 73)( 76, 77)( 79, 80)( 82, 83)( 84, 93)( 85, 95)( 86, 94)( 87, 96)
( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(103,104)(106,107)(109,110)
(111,120)(112,122)(113,121)(114,123)(115,125)(116,124)(117,126)(118,128)
(119,127)(130,131)(133,134)(136,137)(138,147)(139,149)(140,148)(141,150)
(142,152)(143,151)(144,153)(145,155)(146,154)(157,158)(160,161)(163,164);;
s4 := ( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)
( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)( 93,102)
( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)(101,109)
(112,113)(115,116)(118,119)(120,129)(121,131)(122,130)(123,132)(124,134)
(125,133)(126,135)(127,137)(128,136)(139,140)(142,143)(145,146)(147,156)
(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)(155,163);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4*s3*s4, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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, 63)( 31, 65)( 32, 64)
( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)
( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)
(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)
(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)
(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157);
s2 := Sym(164)!( 3,111)( 4,113)( 5,112)( 6,117)( 7,119)( 8,118)( 9,114)
( 10,116)( 11,115)( 12,121)( 13,120)( 14,122)( 15,127)( 16,126)( 17,128)
( 18,124)( 19,123)( 20,125)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)
( 26,135)( 27,134)( 28,133)( 29,132)( 30, 84)( 31, 86)( 32, 85)( 33, 90)
( 34, 92)( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39, 94)( 40, 93)( 41, 95)
( 42,100)( 43, 99)( 44,101)( 45, 97)( 46, 96)( 47, 98)( 48,104)( 49,103)
( 50,102)( 51,110)( 52,109)( 53,108)( 54,107)( 55,106)( 56,105)( 57,144)
( 58,146)( 59,145)( 60,141)( 61,143)( 62,142)( 63,138)( 64,140)( 65,139)
( 66,154)( 67,153)( 68,155)( 69,151)( 70,150)( 71,152)( 72,148)( 73,147)
( 74,149)( 75,164)( 76,163)( 77,162)( 78,161)( 79,160)( 80,159)( 81,158)
( 82,157)( 83,156);
s3 := Sym(164)!( 3, 12)( 4, 14)( 5, 13)( 6, 15)( 7, 17)( 8, 16)( 9, 18)
( 10, 20)( 11, 19)( 22, 23)( 25, 26)( 28, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 42)( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 49, 50)( 52, 53)
( 55, 56)( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)
( 64, 74)( 65, 73)( 76, 77)( 79, 80)( 82, 83)( 84, 93)( 85, 95)( 86, 94)
( 87, 96)( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(103,104)(106,107)
(109,110)(111,120)(112,122)(113,121)(114,123)(115,125)(116,124)(117,126)
(118,128)(119,127)(130,131)(133,134)(136,137)(138,147)(139,149)(140,148)
(141,150)(142,152)(143,151)(144,153)(145,155)(146,154)(157,158)(160,161)
(163,164);
s4 := Sym(164)!( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)
( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 85, 86)( 88, 89)( 91, 92)
( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)
(101,109)(112,113)(115,116)(118,119)(120,129)(121,131)(122,130)(123,132)
(124,134)(125,133)(126,135)(127,137)(128,136)(139,140)(142,143)(145,146)
(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)
(155,163);
poly := sub<Sym(164)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope