Polytope of Type {2,3,6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,18}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 5
Schlafli Type : {2,3,6,18}
Number of vertices, edges, etc : 2, 3, 9, 54, 18
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 : {2,3,6,9}*648
3-fold quotients : {2,3,2,18}*432, {2,3,6,6}*432a
6-fold quotients : {2,3,2,9}*216, {2,3,6,3}*216
9-fold quotients : {2,3,2,6}*144
18-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 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);;
s3 := ( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)( 10, 35)
( 11, 34)( 12, 40)( 13, 39)( 14, 41)( 15, 46)( 16, 45)( 17, 47)( 18, 43)
( 19, 42)( 20, 44)( 21, 50)( 22, 49)( 23, 48)( 24, 56)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 51)( 57, 63)( 58, 65)( 59, 64)( 61, 62)( 66, 73)
( 67, 72)( 68, 74)( 69, 70)( 75, 83)( 76, 82)( 77, 81)( 78, 80)( 84,111)
( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)( 92,115)
( 93,121)( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)(100,123)
(101,125)(102,131)(103,130)(104,129)(105,137)(106,136)(107,135)(108,134)
(109,133)(110,132)(138,144)(139,146)(140,145)(142,143)(147,154)(148,153)
(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);;
s4 := ( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)( 10, 89)
( 11, 88)( 12, 93)( 13, 95)( 14, 94)( 15, 99)( 16,101)( 17,100)( 18, 96)
( 19, 98)( 20, 97)( 21,102)( 22,104)( 23,103)( 24,108)( 25,110)( 26,109)
( 27,105)( 28,107)( 29,106)( 30,144)( 31,146)( 32,145)( 33,141)( 34,143)
( 35,142)( 36,138)( 37,140)( 38,139)( 39,153)( 40,155)( 41,154)( 42,150)
( 43,152)( 44,151)( 45,147)( 46,149)( 47,148)( 48,162)( 49,164)( 50,163)
( 51,159)( 52,161)( 53,160)( 54,156)( 55,158)( 56,157)( 57,117)( 58,119)
( 59,118)( 60,114)( 61,116)( 62,115)( 63,111)( 64,113)( 65,112)( 66,126)
( 67,128)( 68,127)( 69,123)( 70,125)( 71,124)( 72,120)( 73,122)( 74,121)
( 75,135)( 76,137)( 77,136)( 78,132)( 79,134)( 80,133)( 81,129)( 82,131)
( 83,130);;
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,
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(164)!(1,2);
s1 := 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);
s2 := 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);
s3 := Sym(164)!( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)
( 10, 35)( 11, 34)( 12, 40)( 13, 39)( 14, 41)( 15, 46)( 16, 45)( 17, 47)
( 18, 43)( 19, 42)( 20, 44)( 21, 50)( 22, 49)( 23, 48)( 24, 56)( 25, 55)
( 26, 54)( 27, 53)( 28, 52)( 29, 51)( 57, 63)( 58, 65)( 59, 64)( 61, 62)
( 66, 73)( 67, 72)( 68, 74)( 69, 70)( 75, 83)( 76, 82)( 77, 81)( 78, 80)
( 84,111)( 85,113)( 86,112)( 87,117)( 88,119)( 89,118)( 90,114)( 91,116)
( 92,115)( 93,121)( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)
(100,123)(101,125)(102,131)(103,130)(104,129)(105,137)(106,136)(107,135)
(108,134)(109,133)(110,132)(138,144)(139,146)(140,145)(142,143)(147,154)
(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);
s4 := Sym(164)!( 3, 84)( 4, 86)( 5, 85)( 6, 90)( 7, 92)( 8, 91)( 9, 87)
( 10, 89)( 11, 88)( 12, 93)( 13, 95)( 14, 94)( 15, 99)( 16,101)( 17,100)
( 18, 96)( 19, 98)( 20, 97)( 21,102)( 22,104)( 23,103)( 24,108)( 25,110)
( 26,109)( 27,105)( 28,107)( 29,106)( 30,144)( 31,146)( 32,145)( 33,141)
( 34,143)( 35,142)( 36,138)( 37,140)( 38,139)( 39,153)( 40,155)( 41,154)
( 42,150)( 43,152)( 44,151)( 45,147)( 46,149)( 47,148)( 48,162)( 49,164)
( 50,163)( 51,159)( 52,161)( 53,160)( 54,156)( 55,158)( 56,157)( 57,117)
( 58,119)( 59,118)( 60,114)( 61,116)( 62,115)( 63,111)( 64,113)( 65,112)
( 66,126)( 67,128)( 68,127)( 69,123)( 70,125)( 71,124)( 72,120)( 73,122)
( 74,121)( 75,135)( 76,137)( 77,136)( 78,132)( 79,134)( 80,133)( 81,129)
( 82,131)( 83,130);
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, s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope