Polytope of Type {3,2,4,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,18}*1728
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 5
Schlafli Type : {3,2,4,18}
Number of vertices, edges, etc : 3, 3, 8, 72, 36
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 : {3,2,4,9}*864, {3,2,4,18}*864b, {3,2,4,18}*864c
3-fold quotients : {3,2,4,6}*576
4-fold quotients : {3,2,4,9}*432, {3,2,2,18}*432
6-fold quotients : {3,2,4,3}*288, {3,2,4,6}*288b, {3,2,4,6}*288c
8-fold quotients : {3,2,2,9}*216
12-fold quotients : {3,2,4,3}*144, {3,2,2,6}*144
24-fold quotients : {3,2,2,3}*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 := ( 4, 41)( 5, 40)( 6, 43)( 7, 42)( 8, 45)( 9, 44)( 10, 47)( 11, 46)
( 12, 49)( 13, 48)( 14, 51)( 15, 50)( 16, 53)( 17, 52)( 18, 55)( 19, 54)
( 20, 57)( 21, 56)( 22, 59)( 23, 58)( 24, 61)( 25, 60)( 26, 63)( 27, 62)
( 28, 65)( 29, 64)( 30, 67)( 31, 66)( 32, 69)( 33, 68)( 34, 71)( 35, 70)
( 36, 73)( 37, 72)( 38, 75)( 39, 74)( 76,113)( 77,112)( 78,115)( 79,114)
( 80,117)( 81,116)( 82,119)( 83,118)( 84,121)( 85,120)( 86,123)( 87,122)
( 88,125)( 89,124)( 90,127)( 91,126)( 92,129)( 93,128)( 94,131)( 95,130)
( 96,133)( 97,132)( 98,135)( 99,134)(100,137)(101,136)(102,139)(103,138)
(104,141)(105,140)(106,143)(107,142)(108,145)(109,144)(110,147)(111,146);;
s3 := ( 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);;
s4 := ( 4, 88)( 5, 89)( 6, 91)( 7, 90)( 8, 96)( 9, 97)( 10, 99)( 11, 98)
( 12, 92)( 13, 93)( 14, 95)( 15, 94)( 16, 76)( 17, 77)( 18, 79)( 19, 78)
( 20, 84)( 21, 85)( 22, 87)( 23, 86)( 24, 80)( 25, 81)( 26, 83)( 27, 82)
( 28,104)( 29,105)( 30,107)( 31,106)( 32,100)( 33,101)( 34,103)( 35,102)
( 36,108)( 37,109)( 38,111)( 39,110)( 40,124)( 41,125)( 42,127)( 43,126)
( 44,132)( 45,133)( 46,135)( 47,134)( 48,128)( 49,129)( 50,131)( 51,130)
( 52,112)( 53,113)( 54,115)( 55,114)( 56,120)( 57,121)( 58,123)( 59,122)
( 60,116)( 61,117)( 62,119)( 63,118)( 64,140)( 65,141)( 66,143)( 67,142)
( 68,136)( 69,137)( 70,139)( 71,138)( 72,144)( 73,145)( 74,147)( 75,146);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*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(147)!(2,3);
s1 := Sym(147)!(1,2);
s2 := Sym(147)!( 4, 41)( 5, 40)( 6, 43)( 7, 42)( 8, 45)( 9, 44)( 10, 47)
( 11, 46)( 12, 49)( 13, 48)( 14, 51)( 15, 50)( 16, 53)( 17, 52)( 18, 55)
( 19, 54)( 20, 57)( 21, 56)( 22, 59)( 23, 58)( 24, 61)( 25, 60)( 26, 63)
( 27, 62)( 28, 65)( 29, 64)( 30, 67)( 31, 66)( 32, 69)( 33, 68)( 34, 71)
( 35, 70)( 36, 73)( 37, 72)( 38, 75)( 39, 74)( 76,113)( 77,112)( 78,115)
( 79,114)( 80,117)( 81,116)( 82,119)( 83,118)( 84,121)( 85,120)( 86,123)
( 87,122)( 88,125)( 89,124)( 90,127)( 91,126)( 92,129)( 93,128)( 94,131)
( 95,130)( 96,133)( 97,132)( 98,135)( 99,134)(100,137)(101,136)(102,139)
(103,138)(104,141)(105,140)(106,143)(107,142)(108,145)(109,144)(110,147)
(111,146);
s3 := 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);
s4 := Sym(147)!( 4, 88)( 5, 89)( 6, 91)( 7, 90)( 8, 96)( 9, 97)( 10, 99)
( 11, 98)( 12, 92)( 13, 93)( 14, 95)( 15, 94)( 16, 76)( 17, 77)( 18, 79)
( 19, 78)( 20, 84)( 21, 85)( 22, 87)( 23, 86)( 24, 80)( 25, 81)( 26, 83)
( 27, 82)( 28,104)( 29,105)( 30,107)( 31,106)( 32,100)( 33,101)( 34,103)
( 35,102)( 36,108)( 37,109)( 38,111)( 39,110)( 40,124)( 41,125)( 42,127)
( 43,126)( 44,132)( 45,133)( 46,135)( 47,134)( 48,128)( 49,129)( 50,131)
( 51,130)( 52,112)( 53,113)( 54,115)( 55,114)( 56,120)( 57,121)( 58,123)
( 59,122)( 60,116)( 61,117)( 62,119)( 63,118)( 64,140)( 65,141)( 66,143)
( 67,142)( 68,136)( 69,137)( 70,139)( 71,138)( 72,144)( 73,145)( 74,147)
( 75,146);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*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