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