Polytope of Type {10,4,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,18}*1440
Also Known As : {{10,4|2},{4,18|2}}. if this polytope has another name.
Group : SmallGroup(1440,1593)
Rank : 4
Schlafli Type : {10,4,18}
Number of vertices, edges, etc : 10, 20, 36, 18
Order of s0s1s2s3 : 180
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 : {10,2,18}*720
3-fold quotients : {10,4,6}*480
4-fold quotients : {5,2,18}*360, {10,2,9}*360
5-fold quotients : {2,4,18}*288a
6-fold quotients : {10,2,6}*240
8-fold quotients : {5,2,9}*180
9-fold quotients : {10,4,2}*160
10-fold quotients : {2,2,18}*144
12-fold quotients : {5,2,6}*120, {10,2,3}*120
15-fold quotients : {2,4,6}*96a
18-fold quotients : {10,2,2}*80
20-fold quotients : {2,2,9}*72
24-fold quotients : {5,2,3}*60
30-fold quotients : {2,2,6}*48
36-fold quotients : {5,2,2}*40
45-fold quotients : {2,4,2}*32
60-fold quotients : {2,2,3}*24
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 13)( 5, 14)( 6, 15)( 7, 10)( 8, 11)( 9, 12)( 19, 28)( 20, 29)
( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)( 37, 40)
( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)( 54, 57)
( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)( 80, 89)
( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)( 97,100)
( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)
(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)(140,149)
(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)(157,160)
(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177);;
s1 := ( 1, 4)( 2, 5)( 3, 6)( 7, 13)( 8, 14)( 9, 15)( 16, 19)( 17, 20)
( 18, 21)( 22, 28)( 23, 29)( 24, 30)( 31, 34)( 32, 35)( 33, 36)( 37, 43)
( 38, 44)( 39, 45)( 46, 49)( 47, 50)( 48, 51)( 52, 58)( 53, 59)( 54, 60)
( 61, 64)( 62, 65)( 63, 66)( 67, 73)( 68, 74)( 69, 75)( 76, 79)( 77, 80)
( 78, 81)( 82, 88)( 83, 89)( 84, 90)( 91,139)( 92,140)( 93,141)( 94,136)
( 95,137)( 96,138)( 97,148)( 98,149)( 99,150)(100,145)(101,146)(102,147)
(103,142)(104,143)(105,144)(106,154)(107,155)(108,156)(109,151)(110,152)
(111,153)(112,163)(113,164)(114,165)(115,160)(116,161)(117,162)(118,157)
(119,158)(120,159)(121,169)(122,170)(123,171)(124,166)(125,167)(126,168)
(127,178)(128,179)(129,180)(130,175)(131,176)(132,177)(133,172)(134,173)
(135,174);;
s2 := ( 1, 91)( 2, 93)( 3, 92)( 4, 94)( 5, 96)( 6, 95)( 7, 97)( 8, 99)
( 9, 98)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)( 16,123)
( 17,122)( 18,121)( 19,126)( 20,125)( 21,124)( 22,129)( 23,128)( 24,127)
( 25,132)( 26,131)( 27,130)( 28,135)( 29,134)( 30,133)( 31,108)( 32,107)
( 33,106)( 34,111)( 35,110)( 36,109)( 37,114)( 38,113)( 39,112)( 40,117)
( 41,116)( 42,115)( 43,120)( 44,119)( 45,118)( 46,136)( 47,138)( 48,137)
( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)( 56,147)
( 57,146)( 58,148)( 59,150)( 60,149)( 61,168)( 62,167)( 63,166)( 64,171)
( 65,170)( 66,169)( 67,174)( 68,173)( 69,172)( 70,177)( 71,176)( 72,175)
( 73,180)( 74,179)( 75,178)( 76,153)( 77,152)( 78,151)( 79,156)( 80,155)
( 81,154)( 82,159)( 83,158)( 84,157)( 85,162)( 86,161)( 87,160)( 88,165)
( 89,164)( 90,163);;
s3 := ( 1, 16)( 2, 18)( 3, 17)( 4, 19)( 5, 21)( 6, 20)( 7, 22)( 8, 24)
( 9, 23)( 10, 25)( 11, 27)( 12, 26)( 13, 28)( 14, 30)( 15, 29)( 31, 33)
( 34, 36)( 37, 39)( 40, 42)( 43, 45)( 46, 61)( 47, 63)( 48, 62)( 49, 64)
( 50, 66)( 51, 65)( 52, 67)( 53, 69)( 54, 68)( 55, 70)( 56, 72)( 57, 71)
( 58, 73)( 59, 75)( 60, 74)( 76, 78)( 79, 81)( 82, 84)( 85, 87)( 88, 90)
( 91,106)( 92,108)( 93,107)( 94,109)( 95,111)( 96,110)( 97,112)( 98,114)
( 99,113)(100,115)(101,117)(102,116)(103,118)(104,120)(105,119)(121,123)
(124,126)(127,129)(130,132)(133,135)(136,151)(137,153)(138,152)(139,154)
(140,156)(141,155)(142,157)(143,159)(144,158)(145,160)(146,162)(147,161)
(148,163)(149,165)(150,164)(166,168)(169,171)(172,174)(175,177)(178,180);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(180)!( 4, 13)( 5, 14)( 6, 15)( 7, 10)( 8, 11)( 9, 12)( 19, 28)
( 20, 29)( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)
( 37, 40)( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)
( 54, 57)( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)
( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)
( 97,100)( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)
(114,117)(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)
(140,149)(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)
(157,160)(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)
(174,177);
s1 := Sym(180)!( 1, 4)( 2, 5)( 3, 6)( 7, 13)( 8, 14)( 9, 15)( 16, 19)
( 17, 20)( 18, 21)( 22, 28)( 23, 29)( 24, 30)( 31, 34)( 32, 35)( 33, 36)
( 37, 43)( 38, 44)( 39, 45)( 46, 49)( 47, 50)( 48, 51)( 52, 58)( 53, 59)
( 54, 60)( 61, 64)( 62, 65)( 63, 66)( 67, 73)( 68, 74)( 69, 75)( 76, 79)
( 77, 80)( 78, 81)( 82, 88)( 83, 89)( 84, 90)( 91,139)( 92,140)( 93,141)
( 94,136)( 95,137)( 96,138)( 97,148)( 98,149)( 99,150)(100,145)(101,146)
(102,147)(103,142)(104,143)(105,144)(106,154)(107,155)(108,156)(109,151)
(110,152)(111,153)(112,163)(113,164)(114,165)(115,160)(116,161)(117,162)
(118,157)(119,158)(120,159)(121,169)(122,170)(123,171)(124,166)(125,167)
(126,168)(127,178)(128,179)(129,180)(130,175)(131,176)(132,177)(133,172)
(134,173)(135,174);
s2 := Sym(180)!( 1, 91)( 2, 93)( 3, 92)( 4, 94)( 5, 96)( 6, 95)( 7, 97)
( 8, 99)( 9, 98)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)
( 16,123)( 17,122)( 18,121)( 19,126)( 20,125)( 21,124)( 22,129)( 23,128)
( 24,127)( 25,132)( 26,131)( 27,130)( 28,135)( 29,134)( 30,133)( 31,108)
( 32,107)( 33,106)( 34,111)( 35,110)( 36,109)( 37,114)( 38,113)( 39,112)
( 40,117)( 41,116)( 42,115)( 43,120)( 44,119)( 45,118)( 46,136)( 47,138)
( 48,137)( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)
( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,168)( 62,167)( 63,166)
( 64,171)( 65,170)( 66,169)( 67,174)( 68,173)( 69,172)( 70,177)( 71,176)
( 72,175)( 73,180)( 74,179)( 75,178)( 76,153)( 77,152)( 78,151)( 79,156)
( 80,155)( 81,154)( 82,159)( 83,158)( 84,157)( 85,162)( 86,161)( 87,160)
( 88,165)( 89,164)( 90,163);
s3 := Sym(180)!( 1, 16)( 2, 18)( 3, 17)( 4, 19)( 5, 21)( 6, 20)( 7, 22)
( 8, 24)( 9, 23)( 10, 25)( 11, 27)( 12, 26)( 13, 28)( 14, 30)( 15, 29)
( 31, 33)( 34, 36)( 37, 39)( 40, 42)( 43, 45)( 46, 61)( 47, 63)( 48, 62)
( 49, 64)( 50, 66)( 51, 65)( 52, 67)( 53, 69)( 54, 68)( 55, 70)( 56, 72)
( 57, 71)( 58, 73)( 59, 75)( 60, 74)( 76, 78)( 79, 81)( 82, 84)( 85, 87)
( 88, 90)( 91,106)( 92,108)( 93,107)( 94,109)( 95,111)( 96,110)( 97,112)
( 98,114)( 99,113)(100,115)(101,117)(102,116)(103,118)(104,120)(105,119)
(121,123)(124,126)(127,129)(130,132)(133,135)(136,151)(137,153)(138,152)
(139,154)(140,156)(141,155)(142,157)(143,159)(144,158)(145,160)(146,162)
(147,161)(148,163)(149,165)(150,164)(166,168)(169,171)(172,174)(175,177)
(178,180);
poly := sub<Sym(180)|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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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.
to this polytope