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