Polytope of Type {45,10}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope