Polytope of Type {2,10,45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,45}*1800
if this polytope has a name.
Group : SmallGroup(1800,317)
Rank : 4
Schlafli Type : {2,10,45}
Number of vertices, edges, etc : 2, 10, 225, 45
Order of s₀s₁s₂s₃ : 90
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,10,15}*600
   5-fold quotients : {2,2,45}*360
   9-fold quotients : {2,10,5}*200
   15-fold quotients : {2,2,15}*120
   25-fold quotients : {2,2,9}*72
   45-fold quotients : {2,2,5}*40
   75-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 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, 76)( 32, 77)( 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)( 46, 61)( 47, 62)( 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)(106,151)(107,152)(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)(121,136)(122,137)(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)(181,226)(182,227)(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)
(196,211)(197,212);;
s2 := (  3, 18)(  4, 20)(  5, 19)(  6, 30)(  7, 32)(  8, 31)(  9, 27)( 10, 29)
( 11, 28)( 12, 24)( 13, 26)( 14, 25)( 15, 21)( 16, 23)( 17, 22)( 33, 63)
( 34, 65)( 35, 64)( 36, 75)( 37, 77)( 38, 76)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 49, 50)( 51, 60)
( 52, 62)( 53, 61)( 54, 57)( 55, 59)( 56, 58)( 78,170)( 79,169)( 80,168)
( 81,182)( 82,181)( 83,180)( 84,179)( 85,178)( 86,177)( 87,176)( 88,175)
( 89,174)( 90,173)( 91,172)( 92,171)( 93,155)( 94,154)( 95,153)( 96,167)
( 97,166)( 98,165)( 99,164)(100,163)(101,162)(102,161)(103,160)(104,159)
(105,158)(106,157)(107,156)(108,215)(109,214)(110,213)(111,227)(112,226)
(113,225)(114,224)(115,223)(116,222)(117,221)(118,220)(119,219)(120,218)
(121,217)(122,216)(123,200)(124,199)(125,198)(126,212)(127,211)(128,210)
(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)
(137,201)(138,185)(139,184)(140,183)(141,197)(142,196)(143,195)(144,194)
(145,193)(146,192)(147,191)(148,190)(149,189)(150,188)(151,187)(152,186);;
s3 := (  3, 81)(  4, 83)(  5, 82)(  6, 78)(  7, 80)(  8, 79)(  9, 90)( 10, 92)
( 11, 91)( 12, 87)( 13, 89)( 14, 88)( 15, 84)( 16, 86)( 17, 85)( 18,141)
( 19,143)( 20,142)( 21,138)( 22,140)( 23,139)( 24,150)( 25,152)( 26,151)
( 27,147)( 28,149)( 29,148)( 30,144)( 31,146)( 32,145)( 33,126)( 34,128)
( 35,127)( 36,123)( 37,125)( 38,124)( 39,135)( 40,137)( 41,136)( 42,132)
( 43,134)( 44,133)( 45,129)( 46,131)( 47,130)( 48,111)( 49,113)( 50,112)
( 51,108)( 52,110)( 53,109)( 54,120)( 55,122)( 56,121)( 57,117)( 58,119)
( 59,118)( 60,114)( 61,116)( 62,115)( 63, 96)( 64, 98)( 65, 97)( 66, 93)
( 67, 95)( 68, 94)( 69,105)( 70,107)( 71,106)( 72,102)( 73,104)( 74,103)
( 75, 99)( 76,101)( 77,100)(153,158)(154,157)(155,156)(159,167)(160,166)
(161,165)(162,164)(168,218)(169,217)(170,216)(171,215)(172,214)(173,213)
(174,227)(175,226)(176,225)(177,224)(178,223)(179,222)(180,221)(181,220)
(182,219)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,212)
(190,211)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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(227)!(1,2);
s1 := Sym(227)!( 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, 76)( 32, 77)
( 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)( 46, 61)( 47, 62)( 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)(106,151)(107,152)(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)(121,136)(122,137)(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)(181,226)(182,227)(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)(196,211)(197,212);
s2 := Sym(227)!(  3, 18)(  4, 20)(  5, 19)(  6, 30)(  7, 32)(  8, 31)(  9, 27)
( 10, 29)( 11, 28)( 12, 24)( 13, 26)( 14, 25)( 15, 21)( 16, 23)( 17, 22)
( 33, 63)( 34, 65)( 35, 64)( 36, 75)( 37, 77)( 38, 76)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 49, 50)
( 51, 60)( 52, 62)( 53, 61)( 54, 57)( 55, 59)( 56, 58)( 78,170)( 79,169)
( 80,168)( 81,182)( 82,181)( 83,180)( 84,179)( 85,178)( 86,177)( 87,176)
( 88,175)( 89,174)( 90,173)( 91,172)( 92,171)( 93,155)( 94,154)( 95,153)
( 96,167)( 97,166)( 98,165)( 99,164)(100,163)(101,162)(102,161)(103,160)
(104,159)(105,158)(106,157)(107,156)(108,215)(109,214)(110,213)(111,227)
(112,226)(113,225)(114,224)(115,223)(116,222)(117,221)(118,220)(119,219)
(120,218)(121,217)(122,216)(123,200)(124,199)(125,198)(126,212)(127,211)
(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)
(136,202)(137,201)(138,185)(139,184)(140,183)(141,197)(142,196)(143,195)
(144,194)(145,193)(146,192)(147,191)(148,190)(149,189)(150,188)(151,187)
(152,186);
s3 := Sym(227)!(  3, 81)(  4, 83)(  5, 82)(  6, 78)(  7, 80)(  8, 79)(  9, 90)
( 10, 92)( 11, 91)( 12, 87)( 13, 89)( 14, 88)( 15, 84)( 16, 86)( 17, 85)
( 18,141)( 19,143)( 20,142)( 21,138)( 22,140)( 23,139)( 24,150)( 25,152)
( 26,151)( 27,147)( 28,149)( 29,148)( 30,144)( 31,146)( 32,145)( 33,126)
( 34,128)( 35,127)( 36,123)( 37,125)( 38,124)( 39,135)( 40,137)( 41,136)
( 42,132)( 43,134)( 44,133)( 45,129)( 46,131)( 47,130)( 48,111)( 49,113)
( 50,112)( 51,108)( 52,110)( 53,109)( 54,120)( 55,122)( 56,121)( 57,117)
( 58,119)( 59,118)( 60,114)( 61,116)( 62,115)( 63, 96)( 64, 98)( 65, 97)
( 66, 93)( 67, 95)( 68, 94)( 69,105)( 70,107)( 71,106)( 72,102)( 73,104)
( 74,103)( 75, 99)( 76,101)( 77,100)(153,158)(154,157)(155,156)(159,167)
(160,166)(161,165)(162,164)(168,218)(169,217)(170,216)(171,215)(172,214)
(173,213)(174,227)(175,226)(176,225)(177,224)(178,223)(179,222)(180,221)
(181,220)(182,219)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)
(189,212)(190,211)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)
(197,204);
poly := sub<Sym(227)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope