Polytope of Type {6,75}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,75}*900
if this polytope has a name.
Group : SmallGroup(900,37)
Rank : 3
Schlafli Type : {6,75}
Number of vertices, edges, etc : 6, 225, 75
Order of s₀s₁s₂ : 150
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,75,2} of size 1800
Vertex Figure Of :
   {2,6,75} of size 1800
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,75}*300
   5-fold quotients : {6,15}*180
   9-fold quotients : {2,25}*100
   15-fold quotients : {2,15}*60
   25-fold quotients : {6,3}*36
   45-fold quotients : {2,5}*20
   75-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,150}*1800c
Permutation Representation (GAP) :

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