Polytope of Type {10,45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,45}*900
if this polytope has a name.
Group : SmallGroup(900,62)
Rank : 3
Schlafli Type : {10,45}
Number of vertices, edges, etc : 10, 225, 45
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 :
   {10,45,2} of size 1800
Vertex Figure Of :
   {2,10,45} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,15}*300
   5-fold quotients : {2,45}*180
   9-fold quotients : {10,5}*100
   15-fold quotients : {2,15}*60
   25-fold quotients : {2,9}*36
   45-fold quotients : {2,5}*20
   75-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,90}*1800c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope