Polytope of Type {8,10,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,10,5}*800
if this polytope has a name.
Group : SmallGroup(800,703)
Rank : 4
Schlafli Type : {8,10,5}
Number of vertices, edges, etc : 8, 40, 25, 5
Order of s₀s₁s₂s₃ : 40
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,10,5,2} of size 1600
Vertex Figure Of :
   {2,8,10,5} of size 1600
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,10,5}*400
   4-fold quotients : {2,10,5}*200
   5-fold quotients : {8,2,5}*160
   10-fold quotients : {4,2,5}*80
   20-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   2-fold covers : {16,10,5}*1600, {8,10,10}*1600c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope