Polytope of Type {5,10,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10,18}*1800
if this polytope has a name.
Group : SmallGroup(1800,296)
Rank : 4
Schlafli Type : {5,10,18}
Number of vertices, edges, etc : 5, 25, 90, 18
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 : {5,10,6}*600
   5-fold quotients : {5,2,18}*360
   9-fold quotients : {5,10,2}*200
   10-fold quotients : {5,2,9}*180
   15-fold quotients : {5,2,6}*120
   30-fold quotients : {5,2,3}*60
   45-fold quotients : {5,2,2}*40
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)( 16, 61)( 17, 62)
( 18, 63)( 19, 73)( 20, 74)( 21, 75)( 22, 70)( 23, 71)( 24, 72)( 25, 67)
( 26, 68)( 27, 69)( 28, 64)( 29, 65)( 30, 66)( 31, 46)( 32, 47)( 33, 48)
( 34, 58)( 35, 59)( 36, 60)( 37, 55)( 38, 56)( 39, 57)( 40, 52)( 41, 53)
( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 79, 88)( 80, 89)( 81, 90)( 82, 85)
( 83, 86)( 84, 87)( 91,136)( 92,137)( 93,138)( 94,148)( 95,149)( 96,150)
( 97,145)( 98,146)( 99,147)(100,142)(101,143)(102,144)(103,139)(104,140)
(105,141)(106,121)(107,122)(108,123)(109,133)(110,134)(111,135)(112,130)
(113,131)(114,132)(115,127)(116,128)(117,129)(118,124)(119,125)(120,126)
(154,163)(155,164)(156,165)(157,160)(158,161)(159,162)(166,211)(167,212)
(168,213)(169,223)(170,224)(171,225)(172,220)(173,221)(174,222)(175,217)
(176,218)(177,219)(178,214)(179,215)(180,216)(181,196)(182,197)(183,198)
(184,208)(185,209)(186,210)(187,205)(188,206)(189,207)(190,202)(191,203)
(192,204)(193,199)(194,200)(195,201);;
s1 := (  1, 19)(  2, 20)(  3, 21)(  4, 16)(  5, 17)(  6, 18)(  7, 28)(  8, 29)
(  9, 30)( 10, 25)( 11, 26)( 12, 27)( 13, 22)( 14, 23)( 15, 24)( 31, 64)
( 32, 65)( 33, 66)( 34, 61)( 35, 62)( 36, 63)( 37, 73)( 38, 74)( 39, 75)
( 40, 70)( 41, 71)( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 49)( 47, 50)
( 48, 51)( 52, 58)( 53, 59)( 54, 60)( 76, 94)( 77, 95)( 78, 96)( 79, 91)
( 80, 92)( 81, 93)( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)( 87,102)
( 88, 97)( 89, 98)( 90, 99)(106,139)(107,140)(108,141)(109,136)(110,137)
(111,138)(112,148)(113,149)(114,150)(115,145)(116,146)(117,147)(118,142)
(119,143)(120,144)(121,124)(122,125)(123,126)(127,133)(128,134)(129,135)
(151,169)(152,170)(153,171)(154,166)(155,167)(156,168)(157,178)(158,179)
(159,180)(160,175)(161,176)(162,177)(163,172)(164,173)(165,174)(181,214)
(182,215)(183,216)(184,211)(185,212)(186,213)(187,223)(188,224)(189,225)
(190,220)(191,221)(192,222)(193,217)(194,218)(195,219)(196,199)(197,200)
(198,201)(202,208)(203,209)(204,210);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 16, 61)( 17, 63)( 18, 62)
( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)( 25, 70)( 26, 72)
( 27, 71)( 28, 73)( 29, 75)( 30, 74)( 31, 46)( 32, 48)( 33, 47)( 34, 49)
( 35, 51)( 36, 50)( 37, 52)( 38, 54)( 39, 53)( 40, 55)( 41, 57)( 42, 56)
( 43, 58)( 44, 60)( 45, 59)( 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)( 91,212)( 92,211)( 93,213)( 94,215)( 95,214)( 96,216)
( 97,218)( 98,217)( 99,219)(100,221)(101,220)(102,222)(103,224)(104,223)
(105,225)(106,197)(107,196)(108,198)(109,200)(110,199)(111,201)(112,203)
(113,202)(114,204)(115,206)(116,205)(117,207)(118,209)(119,208)(120,210)
(121,182)(122,181)(123,183)(124,185)(125,184)(126,186)(127,188)(128,187)
(129,189)(130,191)(131,190)(132,192)(133,194)(134,193)(135,195)(136,167)
(137,166)(138,168)(139,170)(140,169)(141,171)(142,173)(143,172)(144,174)
(145,176)(146,175)(147,177)(148,179)(149,178)(150,180);;
s3 := (  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, 91)
( 17, 93)( 18, 92)( 19, 94)( 20, 96)( 21, 95)( 22, 97)( 23, 99)( 24, 98)
( 25,100)( 26,102)( 27,101)( 28,103)( 29,105)( 30,104)( 31,106)( 32,108)
( 33,107)( 34,109)( 35,111)( 36,110)( 37,112)( 38,114)( 39,113)( 40,115)
( 41,117)( 42,116)( 43,118)( 44,120)( 45,119)( 46,121)( 47,123)( 48,122)
( 49,124)( 50,126)( 51,125)( 52,127)( 53,129)( 54,128)( 55,130)( 56,132)
( 57,131)( 58,133)( 59,135)( 60,134)( 61,136)( 62,138)( 63,137)( 64,139)
( 65,141)( 66,140)( 67,142)( 68,144)( 69,143)( 70,145)( 71,147)( 72,146)
( 73,148)( 74,150)( 75,149)(151,152)(154,155)(157,158)(160,161)(163,164)
(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)(184,185)(187,188)
(190,191)(193,194)(196,197)(199,200)(202,203)(205,206)(208,209)(211,212)
(214,215)(217,218)(220,221)(223,224);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*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(225)!(  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 16, 61)
( 17, 62)( 18, 63)( 19, 73)( 20, 74)( 21, 75)( 22, 70)( 23, 71)( 24, 72)
( 25, 67)( 26, 68)( 27, 69)( 28, 64)( 29, 65)( 30, 66)( 31, 46)( 32, 47)
( 33, 48)( 34, 58)( 35, 59)( 36, 60)( 37, 55)( 38, 56)( 39, 57)( 40, 52)
( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 79, 88)( 80, 89)( 81, 90)
( 82, 85)( 83, 86)( 84, 87)( 91,136)( 92,137)( 93,138)( 94,148)( 95,149)
( 96,150)( 97,145)( 98,146)( 99,147)(100,142)(101,143)(102,144)(103,139)
(104,140)(105,141)(106,121)(107,122)(108,123)(109,133)(110,134)(111,135)
(112,130)(113,131)(114,132)(115,127)(116,128)(117,129)(118,124)(119,125)
(120,126)(154,163)(155,164)(156,165)(157,160)(158,161)(159,162)(166,211)
(167,212)(168,213)(169,223)(170,224)(171,225)(172,220)(173,221)(174,222)
(175,217)(176,218)(177,219)(178,214)(179,215)(180,216)(181,196)(182,197)
(183,198)(184,208)(185,209)(186,210)(187,205)(188,206)(189,207)(190,202)
(191,203)(192,204)(193,199)(194,200)(195,201);
s1 := Sym(225)!(  1, 19)(  2, 20)(  3, 21)(  4, 16)(  5, 17)(  6, 18)(  7, 28)
(  8, 29)(  9, 30)( 10, 25)( 11, 26)( 12, 27)( 13, 22)( 14, 23)( 15, 24)
( 31, 64)( 32, 65)( 33, 66)( 34, 61)( 35, 62)( 36, 63)( 37, 73)( 38, 74)
( 39, 75)( 40, 70)( 41, 71)( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 49)
( 47, 50)( 48, 51)( 52, 58)( 53, 59)( 54, 60)( 76, 94)( 77, 95)( 78, 96)
( 79, 91)( 80, 92)( 81, 93)( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)
( 87,102)( 88, 97)( 89, 98)( 90, 99)(106,139)(107,140)(108,141)(109,136)
(110,137)(111,138)(112,148)(113,149)(114,150)(115,145)(116,146)(117,147)
(118,142)(119,143)(120,144)(121,124)(122,125)(123,126)(127,133)(128,134)
(129,135)(151,169)(152,170)(153,171)(154,166)(155,167)(156,168)(157,178)
(158,179)(159,180)(160,175)(161,176)(162,177)(163,172)(164,173)(165,174)
(181,214)(182,215)(183,216)(184,211)(185,212)(186,213)(187,223)(188,224)
(189,225)(190,220)(191,221)(192,222)(193,217)(194,218)(195,219)(196,199)
(197,200)(198,201)(202,208)(203,209)(204,210);
s2 := Sym(225)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 16, 61)( 17, 63)
( 18, 62)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)( 25, 70)
( 26, 72)( 27, 71)( 28, 73)( 29, 75)( 30, 74)( 31, 46)( 32, 48)( 33, 47)
( 34, 49)( 35, 51)( 36, 50)( 37, 52)( 38, 54)( 39, 53)( 40, 55)( 41, 57)
( 42, 56)( 43, 58)( 44, 60)( 45, 59)( 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)( 91,212)( 92,211)( 93,213)( 94,215)( 95,214)
( 96,216)( 97,218)( 98,217)( 99,219)(100,221)(101,220)(102,222)(103,224)
(104,223)(105,225)(106,197)(107,196)(108,198)(109,200)(110,199)(111,201)
(112,203)(113,202)(114,204)(115,206)(116,205)(117,207)(118,209)(119,208)
(120,210)(121,182)(122,181)(123,183)(124,185)(125,184)(126,186)(127,188)
(128,187)(129,189)(130,191)(131,190)(132,192)(133,194)(134,193)(135,195)
(136,167)(137,166)(138,168)(139,170)(140,169)(141,171)(142,173)(143,172)
(144,174)(145,176)(146,175)(147,177)(148,179)(149,178)(150,180);
s3 := 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, 91)( 17, 93)( 18, 92)( 19, 94)( 20, 96)( 21, 95)( 22, 97)( 23, 99)
( 24, 98)( 25,100)( 26,102)( 27,101)( 28,103)( 29,105)( 30,104)( 31,106)
( 32,108)( 33,107)( 34,109)( 35,111)( 36,110)( 37,112)( 38,114)( 39,113)
( 40,115)( 41,117)( 42,116)( 43,118)( 44,120)( 45,119)( 46,121)( 47,123)
( 48,122)( 49,124)( 50,126)( 51,125)( 52,127)( 53,129)( 54,128)( 55,130)
( 56,132)( 57,131)( 58,133)( 59,135)( 60,134)( 61,136)( 62,138)( 63,137)
( 64,139)( 65,141)( 66,140)( 67,142)( 68,144)( 69,143)( 70,145)( 71,147)
( 72,146)( 73,148)( 74,150)( 75,149)(151,152)(154,155)(157,158)(160,161)
(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)(184,185)
(187,188)(190,191)(193,194)(196,197)(199,200)(202,203)(205,206)(208,209)
(211,212)(214,215)(217,218)(220,221)(223,224);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*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