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