Questions?
See the FAQ
or other info.

Polytope of Type {18,10,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,10,5}*1800
if this polytope has a name.
Group : SmallGroup(1800,296)
Rank : 4
Schlafli Type : {18,10,5}
Number of vertices, edges, etc : 18, 90, 25, 5
Order of s0s1s2s3 : 90
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) :
   3-fold quotients : {6,10,5}*600
   5-fold quotients : {18,2,5}*360
   9-fold quotients : {2,10,5}*200
   10-fold quotients : {9,2,5}*180
   15-fold quotients : {6,2,5}*120
   30-fold quotients : {3,2,5}*60
   45-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 76,152)( 77,151)( 78,153)( 79,155)( 80,154)( 81,156)( 82,158)
( 83,157)( 84,159)( 85,161)( 86,160)( 87,162)( 88,164)( 89,163)( 90,165)
( 91,167)( 92,166)( 93,168)( 94,170)( 95,169)( 96,171)( 97,173)( 98,172)
( 99,174)(100,176)(101,175)(102,177)(103,179)(104,178)(105,180)(106,182)
(107,181)(108,183)(109,185)(110,184)(111,186)(112,188)(113,187)(114,189)
(115,191)(116,190)(117,192)(118,194)(119,193)(120,195)(121,197)(122,196)
(123,198)(124,200)(125,199)(126,201)(127,203)(128,202)(129,204)(130,206)
(131,205)(132,207)(133,209)(134,208)(135,210)(136,212)(137,211)(138,213)
(139,215)(140,214)(141,216)(142,218)(143,217)(144,219)(145,221)(146,220)
(147,222)(148,224)(149,223)(150,225);;
s1 := (  1, 76)(  2, 78)(  3, 77)(  4, 79)(  5, 81)(  6, 80)(  7, 82)(  8, 84)
(  9, 83)( 10, 85)( 11, 87)( 12, 86)( 13, 88)( 14, 90)( 15, 89)( 16,136)
( 17,138)( 18,137)( 19,139)( 20,141)( 21,140)( 22,142)( 23,144)( 24,143)
( 25,145)( 26,147)( 27,146)( 28,148)( 29,150)( 30,149)( 31,121)( 32,123)
( 33,122)( 34,124)( 35,126)( 36,125)( 37,127)( 38,129)( 39,128)( 40,130)
( 41,132)( 42,131)( 43,133)( 44,135)( 45,134)( 46,106)( 47,108)( 48,107)
( 49,109)( 50,111)( 51,110)( 52,112)( 53,114)( 54,113)( 55,115)( 56,117)
( 57,116)( 58,118)( 59,120)( 60,119)( 61, 91)( 62, 93)( 63, 92)( 64, 94)
( 65, 96)( 66, 95)( 67, 97)( 68, 99)( 69, 98)( 70,100)( 71,102)( 72,101)
( 73,103)( 74,105)( 75,104)(151,152)(154,155)(157,158)(160,161)(163,164)
(166,212)(167,211)(168,213)(169,215)(170,214)(171,216)(172,218)(173,217)
(174,219)(175,221)(176,220)(177,222)(178,224)(179,223)(180,225)(181,197)
(182,196)(183,198)(184,200)(185,199)(186,201)(187,203)(188,202)(189,204)
(190,206)(191,205)(192,207)(193,209)(194,208)(195,210);;
s2 := (  1, 16)(  2, 17)(  3, 18)(  4, 28)(  5, 29)(  6, 30)(  7, 25)(  8, 26)
(  9, 27)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)( 15, 21)( 31, 61)
( 32, 62)( 33, 63)( 34, 73)( 35, 74)( 36, 75)( 37, 70)( 38, 71)( 39, 72)
( 40, 67)( 41, 68)( 42, 69)( 43, 64)( 44, 65)( 45, 66)( 49, 58)( 50, 59)
( 51, 60)( 52, 55)( 53, 56)( 54, 57)( 76, 91)( 77, 92)( 78, 93)( 79,103)
( 80,104)( 81,105)( 82,100)( 83,101)( 84,102)( 85, 97)( 86, 98)( 87, 99)
( 88, 94)( 89, 95)( 90, 96)(106,136)(107,137)(108,138)(109,148)(110,149)
(111,150)(112,145)(113,146)(114,147)(115,142)(116,143)(117,144)(118,139)
(119,140)(120,141)(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)
(151,166)(152,167)(153,168)(154,178)(155,179)(156,180)(157,175)(158,176)
(159,177)(160,172)(161,173)(162,174)(163,169)(164,170)(165,171)(181,211)
(182,212)(183,213)(184,223)(185,224)(186,225)(187,220)(188,221)(189,222)
(190,217)(191,218)(192,219)(193,214)(194,215)(195,216)(199,208)(200,209)
(201,210)(202,205)(203,206)(204,207);;
s3 := (  1,  4)(  2,  5)(  3,  6)(  7, 13)(  8, 14)(  9, 15)( 16, 64)( 17, 65)
( 18, 66)( 19, 61)( 20, 62)( 21, 63)( 22, 73)( 23, 74)( 24, 75)( 25, 70)
( 26, 71)( 27, 72)( 28, 67)( 29, 68)( 30, 69)( 31, 49)( 32, 50)( 33, 51)
( 34, 46)( 35, 47)( 36, 48)( 37, 58)( 38, 59)( 39, 60)( 40, 55)( 41, 56)
( 42, 57)( 43, 52)( 44, 53)( 45, 54)( 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)(106,124)(107,125)(108,126)(109,121)(110,122)(111,123)(112,133)
(113,134)(114,135)(115,130)(116,131)(117,132)(118,127)(119,128)(120,129)
(151,154)(152,155)(153,156)(157,163)(158,164)(159,165)(166,214)(167,215)
(168,216)(169,211)(170,212)(171,213)(172,223)(173,224)(174,225)(175,220)
(176,221)(177,222)(178,217)(179,218)(180,219)(181,199)(182,200)(183,201)
(184,196)(185,197)(186,198)(187,208)(188,209)(189,210)(190,205)(191,206)
(192,207)(193,202)(194,203)(195,204);;
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, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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(225)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)
( 71, 72)( 74, 75)( 76,152)( 77,151)( 78,153)( 79,155)( 80,154)( 81,156)
( 82,158)( 83,157)( 84,159)( 85,161)( 86,160)( 87,162)( 88,164)( 89,163)
( 90,165)( 91,167)( 92,166)( 93,168)( 94,170)( 95,169)( 96,171)( 97,173)
( 98,172)( 99,174)(100,176)(101,175)(102,177)(103,179)(104,178)(105,180)
(106,182)(107,181)(108,183)(109,185)(110,184)(111,186)(112,188)(113,187)
(114,189)(115,191)(116,190)(117,192)(118,194)(119,193)(120,195)(121,197)
(122,196)(123,198)(124,200)(125,199)(126,201)(127,203)(128,202)(129,204)
(130,206)(131,205)(132,207)(133,209)(134,208)(135,210)(136,212)(137,211)
(138,213)(139,215)(140,214)(141,216)(142,218)(143,217)(144,219)(145,221)
(146,220)(147,222)(148,224)(149,223)(150,225);
s1 := Sym(225)!(  1, 76)(  2, 78)(  3, 77)(  4, 79)(  5, 81)(  6, 80)(  7, 82)
(  8, 84)(  9, 83)( 10, 85)( 11, 87)( 12, 86)( 13, 88)( 14, 90)( 15, 89)
( 16,136)( 17,138)( 18,137)( 19,139)( 20,141)( 21,140)( 22,142)( 23,144)
( 24,143)( 25,145)( 26,147)( 27,146)( 28,148)( 29,150)( 30,149)( 31,121)
( 32,123)( 33,122)( 34,124)( 35,126)( 36,125)( 37,127)( 38,129)( 39,128)
( 40,130)( 41,132)( 42,131)( 43,133)( 44,135)( 45,134)( 46,106)( 47,108)
( 48,107)( 49,109)( 50,111)( 51,110)( 52,112)( 53,114)( 54,113)( 55,115)
( 56,117)( 57,116)( 58,118)( 59,120)( 60,119)( 61, 91)( 62, 93)( 63, 92)
( 64, 94)( 65, 96)( 66, 95)( 67, 97)( 68, 99)( 69, 98)( 70,100)( 71,102)
( 72,101)( 73,103)( 74,105)( 75,104)(151,152)(154,155)(157,158)(160,161)
(163,164)(166,212)(167,211)(168,213)(169,215)(170,214)(171,216)(172,218)
(173,217)(174,219)(175,221)(176,220)(177,222)(178,224)(179,223)(180,225)
(181,197)(182,196)(183,198)(184,200)(185,199)(186,201)(187,203)(188,202)
(189,204)(190,206)(191,205)(192,207)(193,209)(194,208)(195,210);
s2 := Sym(225)!(  1, 16)(  2, 17)(  3, 18)(  4, 28)(  5, 29)(  6, 30)(  7, 25)
(  8, 26)(  9, 27)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)( 15, 21)
( 31, 61)( 32, 62)( 33, 63)( 34, 73)( 35, 74)( 36, 75)( 37, 70)( 38, 71)
( 39, 72)( 40, 67)( 41, 68)( 42, 69)( 43, 64)( 44, 65)( 45, 66)( 49, 58)
( 50, 59)( 51, 60)( 52, 55)( 53, 56)( 54, 57)( 76, 91)( 77, 92)( 78, 93)
( 79,103)( 80,104)( 81,105)( 82,100)( 83,101)( 84,102)( 85, 97)( 86, 98)
( 87, 99)( 88, 94)( 89, 95)( 90, 96)(106,136)(107,137)(108,138)(109,148)
(110,149)(111,150)(112,145)(113,146)(114,147)(115,142)(116,143)(117,144)
(118,139)(119,140)(120,141)(124,133)(125,134)(126,135)(127,130)(128,131)
(129,132)(151,166)(152,167)(153,168)(154,178)(155,179)(156,180)(157,175)
(158,176)(159,177)(160,172)(161,173)(162,174)(163,169)(164,170)(165,171)
(181,211)(182,212)(183,213)(184,223)(185,224)(186,225)(187,220)(188,221)
(189,222)(190,217)(191,218)(192,219)(193,214)(194,215)(195,216)(199,208)
(200,209)(201,210)(202,205)(203,206)(204,207);
s3 := Sym(225)!(  1,  4)(  2,  5)(  3,  6)(  7, 13)(  8, 14)(  9, 15)( 16, 64)
( 17, 65)( 18, 66)( 19, 61)( 20, 62)( 21, 63)( 22, 73)( 23, 74)( 24, 75)
( 25, 70)( 26, 71)( 27, 72)( 28, 67)( 29, 68)( 30, 69)( 31, 49)( 32, 50)
( 33, 51)( 34, 46)( 35, 47)( 36, 48)( 37, 58)( 38, 59)( 39, 60)( 40, 55)
( 41, 56)( 42, 57)( 43, 52)( 44, 53)( 45, 54)( 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)(106,124)(107,125)(108,126)(109,121)(110,122)(111,123)
(112,133)(113,134)(114,135)(115,130)(116,131)(117,132)(118,127)(119,128)
(120,129)(151,154)(152,155)(153,156)(157,163)(158,164)(159,165)(166,214)
(167,215)(168,216)(169,211)(170,212)(171,213)(172,223)(173,224)(174,225)
(175,220)(176,221)(177,222)(178,217)(179,218)(180,219)(181,199)(182,200)
(183,201)(184,196)(185,197)(186,198)(187,208)(188,209)(189,210)(190,205)
(191,206)(192,207)(193,202)(194,203)(195,204);
poly := sub<Sym(225)|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, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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