Polytope of Type {18,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4,4}*1152a
Also Known As : {{18,4|2},{4,4}4}. if this polytope has another name.
Group : SmallGroup(1152,99252)
Rank : 4
Schlafli Type : {18,4,4}
Number of vertices, edges, etc : 18, 72, 16, 8
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {18,4,4}*576
3-fold quotients : {6,4,4}*384a
4-fold quotients : {18,2,4}*288, {18,4,2}*288a
6-fold quotients : {6,4,4}*192
8-fold quotients : {9,2,4}*144, {18,2,2}*144
9-fold quotients : {2,4,4}*128
12-fold quotients : {6,2,4}*96, {6,4,2}*96a
16-fold quotients : {9,2,2}*72
18-fold quotients : {2,4,4}*64
24-fold quotients : {3,2,4}*48, {6,2,2}*48
36-fold quotients : {2,2,4}*32, {2,4,2}*32
48-fold quotients : {3,2,2}*24
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 37)( 2, 39)( 3, 38)( 4, 45)( 5, 44)( 6, 43)( 7, 42)( 8, 41)
( 9, 40)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 55)( 20, 57)( 21, 56)( 22, 63)( 23, 62)( 24, 61)
( 25, 60)( 26, 59)( 27, 58)( 28, 64)( 29, 66)( 30, 65)( 31, 72)( 32, 71)
( 33, 70)( 34, 69)( 35, 68)( 36, 67)( 73,109)( 74,111)( 75,110)( 76,117)
( 77,116)( 78,115)( 79,114)( 80,113)( 81,112)( 82,118)( 83,120)( 84,119)
( 85,126)( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)( 91,127)( 92,129)
( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)( 98,131)( 99,130)(100,136)
(101,138)(102,137)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139);;
s1 := ( 1, 40)( 2, 42)( 3, 41)( 4, 37)( 5, 39)( 6, 38)( 7, 45)( 8, 44)
( 9, 43)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)( 16, 54)
( 17, 53)( 18, 52)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)( 24, 65)
( 25, 72)( 26, 71)( 27, 70)( 28, 58)( 29, 60)( 30, 59)( 31, 55)( 32, 57)
( 33, 56)( 34, 63)( 35, 62)( 36, 61)( 73,112)( 74,114)( 75,113)( 76,109)
( 77,111)( 78,110)( 79,117)( 80,116)( 81,115)( 82,121)( 83,123)( 84,122)
( 85,118)( 86,120)( 87,119)( 88,126)( 89,125)( 90,124)( 91,139)( 92,141)
( 93,140)( 94,136)( 95,138)( 96,137)( 97,144)( 98,143)( 99,142)(100,130)
(101,132)(102,131)(103,127)(104,129)(105,128)(106,135)(107,134)(108,133);;
s2 := ( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)( 79, 97)( 80, 98)
( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)( 87,105)( 88,106)
( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)
(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)
(123,141)(124,142)(125,143)(126,144);;
s3 := ( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 77)( 6, 78)( 7, 79)( 8, 80)
( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(144)!( 1, 37)( 2, 39)( 3, 38)( 4, 45)( 5, 44)( 6, 43)( 7, 42)
( 8, 41)( 9, 40)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 55)( 20, 57)( 21, 56)( 22, 63)( 23, 62)
( 24, 61)( 25, 60)( 26, 59)( 27, 58)( 28, 64)( 29, 66)( 30, 65)( 31, 72)
( 32, 71)( 33, 70)( 34, 69)( 35, 68)( 36, 67)( 73,109)( 74,111)( 75,110)
( 76,117)( 77,116)( 78,115)( 79,114)( 80,113)( 81,112)( 82,118)( 83,120)
( 84,119)( 85,126)( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)( 91,127)
( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)( 98,131)( 99,130)
(100,136)(101,138)(102,137)(103,144)(104,143)(105,142)(106,141)(107,140)
(108,139);
s1 := Sym(144)!( 1, 40)( 2, 42)( 3, 41)( 4, 37)( 5, 39)( 6, 38)( 7, 45)
( 8, 44)( 9, 43)( 10, 49)( 11, 51)( 12, 50)( 13, 46)( 14, 48)( 15, 47)
( 16, 54)( 17, 53)( 18, 52)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)
( 24, 65)( 25, 72)( 26, 71)( 27, 70)( 28, 58)( 29, 60)( 30, 59)( 31, 55)
( 32, 57)( 33, 56)( 34, 63)( 35, 62)( 36, 61)( 73,112)( 74,114)( 75,113)
( 76,109)( 77,111)( 78,110)( 79,117)( 80,116)( 81,115)( 82,121)( 83,123)
( 84,122)( 85,118)( 86,120)( 87,119)( 88,126)( 89,125)( 90,124)( 91,139)
( 92,141)( 93,140)( 94,136)( 95,138)( 96,137)( 97,144)( 98,143)( 99,142)
(100,130)(101,132)(102,131)(103,127)(104,129)(105,128)(106,135)(107,134)
(108,133);
s2 := Sym(144)!( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)( 79, 97)
( 80, 98)( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)( 87,105)
( 88,106)( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)(113,131)
(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)
(122,140)(123,141)(124,142)(125,143)(126,144);
s3 := Sym(144)!( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 77)( 6, 78)( 7, 79)
( 8, 80)( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)
( 24, 96)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)
( 32,104)( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)
( 40,112)( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)
( 48,120)( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)
( 56,128)( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)
( 64,136)( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)
( 72,144);
poly := sub<Sym(144)|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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope