Polytope of Type {3,2,18,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,18,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {3,2,18,4}
Number of vertices, edges, etc : 3, 3, 36, 72, 8
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 : {3,2,9,4}*864, {3,2,18,4}*864b, {3,2,18,4}*864c
3-fold quotients : {3,2,6,4}*576
4-fold quotients : {3,2,9,4}*432, {3,2,18,2}*432
6-fold quotients : {3,2,3,4}*288, {3,2,6,4}*288b, {3,2,6,4}*288c
8-fold quotients : {3,2,9,2}*216
12-fold quotients : {3,2,3,4}*144, {3,2,6,2}*144
24-fold quotients : {3,2,3,2}*72
36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 8, 12)( 9, 14)( 10, 13)( 11, 15)( 16, 32)( 17, 34)( 18, 33)
( 19, 35)( 20, 28)( 21, 30)( 22, 29)( 23, 31)( 24, 36)( 25, 38)( 26, 37)
( 27, 39)( 41, 42)( 44, 48)( 45, 50)( 46, 49)( 47, 51)( 52, 68)( 53, 70)
( 54, 69)( 55, 71)( 56, 64)( 57, 66)( 58, 65)( 59, 67)( 60, 72)( 61, 74)
( 62, 73)( 63, 75)( 77, 78)( 80, 84)( 81, 86)( 82, 85)( 83, 87)( 88,104)
( 89,106)( 90,105)( 91,107)( 92,100)( 93,102)( 94,101)( 95,103)( 96,108)
( 97,110)( 98,109)( 99,111)(113,114)(116,120)(117,122)(118,121)(119,123)
(124,140)(125,142)(126,141)(127,143)(128,136)(129,138)(130,137)(131,139)
(132,144)(133,146)(134,145)(135,147);;
s3 := ( 4, 52)( 5, 53)( 6, 55)( 7, 54)( 8, 60)( 9, 61)( 10, 63)( 11, 62)
( 12, 56)( 13, 57)( 14, 59)( 15, 58)( 16, 40)( 17, 41)( 18, 43)( 19, 42)
( 20, 48)( 21, 49)( 22, 51)( 23, 50)( 24, 44)( 25, 45)( 26, 47)( 27, 46)
( 28, 68)( 29, 69)( 30, 71)( 31, 70)( 32, 64)( 33, 65)( 34, 67)( 35, 66)
( 36, 72)( 37, 73)( 38, 75)( 39, 74)( 76,124)( 77,125)( 78,127)( 79,126)
( 80,132)( 81,133)( 82,135)( 83,134)( 84,128)( 85,129)( 86,131)( 87,130)
( 88,112)( 89,113)( 90,115)( 91,114)( 92,120)( 93,121)( 94,123)( 95,122)
( 96,116)( 97,117)( 98,119)( 99,118)(100,140)(101,141)(102,143)(103,142)
(104,136)(105,137)(106,139)(107,138)(108,144)(109,145)(110,147)(111,146);;
s4 := ( 4, 79)( 5, 78)( 6, 77)( 7, 76)( 8, 83)( 9, 82)( 10, 81)( 11, 80)
( 12, 87)( 13, 86)( 14, 85)( 15, 84)( 16, 91)( 17, 90)( 18, 89)( 19, 88)
( 20, 95)( 21, 94)( 22, 93)( 23, 92)( 24, 99)( 25, 98)( 26, 97)( 27, 96)
( 28,103)( 29,102)( 30,101)( 31,100)( 32,107)( 33,106)( 34,105)( 35,104)
( 36,111)( 37,110)( 38,109)( 39,108)( 40,115)( 41,114)( 42,113)( 43,112)
( 44,119)( 45,118)( 46,117)( 47,116)( 48,123)( 49,122)( 50,121)( 51,120)
( 52,127)( 53,126)( 54,125)( 55,124)( 56,131)( 57,130)( 58,129)( 59,128)
( 60,135)( 61,134)( 62,133)( 63,132)( 64,139)( 65,138)( 66,137)( 67,136)
( 68,143)( 69,142)( 70,141)( 71,140)( 72,147)( 73,146)( 74,145)( 75,144);;
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*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(147)!(2,3);
s1 := Sym(147)!(1,2);
s2 := Sym(147)!( 5, 6)( 8, 12)( 9, 14)( 10, 13)( 11, 15)( 16, 32)( 17, 34)
( 18, 33)( 19, 35)( 20, 28)( 21, 30)( 22, 29)( 23, 31)( 24, 36)( 25, 38)
( 26, 37)( 27, 39)( 41, 42)( 44, 48)( 45, 50)( 46, 49)( 47, 51)( 52, 68)
( 53, 70)( 54, 69)( 55, 71)( 56, 64)( 57, 66)( 58, 65)( 59, 67)( 60, 72)
( 61, 74)( 62, 73)( 63, 75)( 77, 78)( 80, 84)( 81, 86)( 82, 85)( 83, 87)
( 88,104)( 89,106)( 90,105)( 91,107)( 92,100)( 93,102)( 94,101)( 95,103)
( 96,108)( 97,110)( 98,109)( 99,111)(113,114)(116,120)(117,122)(118,121)
(119,123)(124,140)(125,142)(126,141)(127,143)(128,136)(129,138)(130,137)
(131,139)(132,144)(133,146)(134,145)(135,147);
s3 := Sym(147)!( 4, 52)( 5, 53)( 6, 55)( 7, 54)( 8, 60)( 9, 61)( 10, 63)
( 11, 62)( 12, 56)( 13, 57)( 14, 59)( 15, 58)( 16, 40)( 17, 41)( 18, 43)
( 19, 42)( 20, 48)( 21, 49)( 22, 51)( 23, 50)( 24, 44)( 25, 45)( 26, 47)
( 27, 46)( 28, 68)( 29, 69)( 30, 71)( 31, 70)( 32, 64)( 33, 65)( 34, 67)
( 35, 66)( 36, 72)( 37, 73)( 38, 75)( 39, 74)( 76,124)( 77,125)( 78,127)
( 79,126)( 80,132)( 81,133)( 82,135)( 83,134)( 84,128)( 85,129)( 86,131)
( 87,130)( 88,112)( 89,113)( 90,115)( 91,114)( 92,120)( 93,121)( 94,123)
( 95,122)( 96,116)( 97,117)( 98,119)( 99,118)(100,140)(101,141)(102,143)
(103,142)(104,136)(105,137)(106,139)(107,138)(108,144)(109,145)(110,147)
(111,146);
s4 := Sym(147)!( 4, 79)( 5, 78)( 6, 77)( 7, 76)( 8, 83)( 9, 82)( 10, 81)
( 11, 80)( 12, 87)( 13, 86)( 14, 85)( 15, 84)( 16, 91)( 17, 90)( 18, 89)
( 19, 88)( 20, 95)( 21, 94)( 22, 93)( 23, 92)( 24, 99)( 25, 98)( 26, 97)
( 27, 96)( 28,103)( 29,102)( 30,101)( 31,100)( 32,107)( 33,106)( 34,105)
( 35,104)( 36,111)( 37,110)( 38,109)( 39,108)( 40,115)( 41,114)( 42,113)
( 43,112)( 44,119)( 45,118)( 46,117)( 47,116)( 48,123)( 49,122)( 50,121)
( 51,120)( 52,127)( 53,126)( 54,125)( 55,124)( 56,131)( 57,130)( 58,129)
( 59,128)( 60,135)( 61,134)( 62,133)( 63,132)( 64,139)( 65,138)( 66,137)
( 67,136)( 68,143)( 69,142)( 70,141)( 71,140)( 72,147)( 73,146)( 74,145)
( 75,144);
poly := sub<Sym(147)|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*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope