Polytope of Type {10,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,18}*900
if this polytope has a name.
Group : SmallGroup(900,48)
Rank : 3
Schlafli Type : {10,18}
Number of vertices, edges, etc : 25, 225, 45
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,18,2} of size 1800
Vertex Figure Of :
   {2,10,18} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,6}*300
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,18}*1800b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope