Polytope of Type {10,90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,90}*1800a
if this polytope has a name.
Group : SmallGroup(1800,296)
Rank : 3
Schlafli Type : {10,90}
Number of vertices, edges, etc : 10, 450, 90
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {10,30}*600a
   5-fold quotients : {10,18}*360
   9-fold quotients : {10,10}*200c
   15-fold quotients : {10,6}*120
   18-fold quotients : {5,10}*100
   25-fold quotients : {2,18}*72
   45-fold quotients : {10,2}*40
   50-fold quotients : {2,9}*36
   75-fold quotients : {2,6}*24
   90-fold quotients : {5,2}*20
   150-fold quotients : {2,3}*12
   225-fold quotients : {2,2}*8
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, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)(  8, 30)
(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)( 31, 64)
( 32, 66)( 33, 65)( 34, 61)( 35, 63)( 36, 62)( 37, 73)( 38, 75)( 39, 74)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 49)( 47, 51)
( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 76,170)( 77,169)( 78,171)
( 79,167)( 80,166)( 81,168)( 82,179)( 83,178)( 84,180)( 85,176)( 86,175)
( 87,177)( 88,173)( 89,172)( 90,174)( 91,155)( 92,154)( 93,156)( 94,152)
( 95,151)( 96,153)( 97,164)( 98,163)( 99,165)(100,161)(101,160)(102,162)
(103,158)(104,157)(105,159)(106,215)(107,214)(108,216)(109,212)(110,211)
(111,213)(112,224)(113,223)(114,225)(115,221)(116,220)(117,222)(118,218)
(119,217)(120,219)(121,200)(122,199)(123,201)(124,197)(125,196)(126,198)
(127,209)(128,208)(129,210)(130,206)(131,205)(132,207)(133,203)(134,202)
(135,204)(136,185)(137,184)(138,186)(139,182)(140,181)(141,183)(142,194)
(143,193)(144,195)(145,191)(146,190)(147,192)(148,188)(149,187)(150,189);;
s2 := (  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);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*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*s0*s1*s2*s1*s2*s1*s2*s1 >;

References : None.
to this polytope