Polytope of Type {2,75,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,75,6}*1800
if this polytope has a name.
Group : SmallGroup(1800,246)
Rank : 4
Schlafli Type : {2,75,6}
Number of vertices, edges, etc : 2, 75, 225, 6
Order of s₀s₁s₂s₃ : 150
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 : {2,75,2}*600
   5-fold quotients : {2,15,6}*360
   9-fold quotients : {2,25,2}*200
   15-fold quotients : {2,15,2}*120
   25-fold quotients : {2,3,6}*72
   45-fold quotients : {2,5,2}*40
   75-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope