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