Polytope of Type {90,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,10}*1800a
if this polytope has a name.
Group : SmallGroup(1800,296)
Rank : 3
Schlafli Type : {90,10}
Number of vertices, edges, etc : 90, 450, 10
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {30,10}*600a
   5-fold quotients : {18,10}*360
   9-fold quotients : {10,10}*200b
   15-fold quotients : {6,10}*120
   18-fold quotients : {10,5}*100
   25-fold quotients : {18,2}*72
   45-fold quotients : {2,10}*40
   50-fold quotients : {9,2}*36
   75-fold quotients : {6,2}*24
   90-fold quotients : {2,5}*20
   150-fold quotients : {3,2}*12
   225-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

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

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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*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