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Polytope of Type {10,20,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,20,4}*1600a
Also Known As : {{10,20|2},{20,4|2}}. if this polytope has another name.
Group : SmallGroup(1600,7723)
Rank : 4
Schlafli Type : {10,20,4}
Number of vertices, edges, etc : 10, 100, 40, 4
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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) :
   2-fold quotients : {10,20,2}*800a, {10,10,4}*800a
   4-fold quotients : {10,10,2}*400a
   5-fold quotients : {2,20,4}*320, {10,4,4}*320
   10-fold quotients : {2,20,2}*160, {2,10,4}*160, {10,2,4}*160, {10,4,2}*160
   20-fold quotients : {5,2,4}*80, {2,10,2}*80, {10,2,2}*80
   25-fold quotients : {2,4,4}*64
   40-fold quotients : {2,5,2}*40, {5,2,2}*40
   50-fold quotients : {2,2,4}*32, {2,4,2}*32
   100-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)( 13, 18)
( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 41)
( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)( 59, 74)
( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)( 82, 97)
( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)( 90, 95)
(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)(113,118)
(114,119)(115,120)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)
(137,142)(138,143)(139,144)(140,145)(156,171)(157,172)(158,173)(159,174)
(160,175)(161,166)(162,167)(163,168)(164,169)(165,170)(181,196)(182,197)
(183,198)(184,199)(185,200)(186,191)(187,192)(188,193)(189,194)(190,195);;
s1 := (  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)
(101,131)(102,135)(103,134)(104,133)(105,132)(106,126)(107,130)(108,129)
(109,128)(110,127)(111,146)(112,150)(113,149)(114,148)(115,147)(116,141)
(117,145)(118,144)(119,143)(120,142)(121,136)(122,140)(123,139)(124,138)
(125,137)(151,181)(152,185)(153,184)(154,183)(155,182)(156,176)(157,180)
(158,179)(159,178)(160,177)(161,196)(162,200)(163,199)(164,198)(165,197)
(166,191)(167,195)(168,194)(169,193)(170,192)(171,186)(172,190)(173,189)
(174,188)(175,187);;
s2 := (  1,102)(  2,101)(  3,105)(  4,104)(  5,103)(  6,107)(  7,106)(  8,110)
(  9,109)( 10,108)( 11,112)( 12,111)( 13,115)( 14,114)( 15,113)( 16,117)
( 17,116)( 18,120)( 19,119)( 20,118)( 21,122)( 22,121)( 23,125)( 24,124)
( 25,123)( 26,127)( 27,126)( 28,130)( 29,129)( 30,128)( 31,132)( 32,131)
( 33,135)( 34,134)( 35,133)( 36,137)( 37,136)( 38,140)( 39,139)( 40,138)
( 41,142)( 42,141)( 43,145)( 44,144)( 45,143)( 46,147)( 47,146)( 48,150)
( 49,149)( 50,148)( 51,152)( 52,151)( 53,155)( 54,154)( 55,153)( 56,157)
( 57,156)( 58,160)( 59,159)( 60,158)( 61,162)( 62,161)( 63,165)( 64,164)
( 65,163)( 66,167)( 67,166)( 68,170)( 69,169)( 70,168)( 71,172)( 72,171)
( 73,175)( 74,174)( 75,173)( 76,177)( 77,176)( 78,180)( 79,179)( 80,178)
( 81,182)( 82,181)( 83,185)( 84,184)( 85,183)( 86,187)( 87,186)( 88,190)
( 89,189)( 90,188)( 91,192)( 92,191)( 93,195)( 94,194)( 95,193)( 96,197)
( 97,196)( 98,200)( 99,199)(100,198);;
s3 := (101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)
(109,159)(110,160)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)
(117,167)(118,168)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)
(125,175)(126,176)(127,177)(128,178)(129,179)(130,180)(131,181)(132,182)
(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)
(141,191)(142,192)(143,193)(144,194)(145,195)(146,196)(147,197)(148,198)
(149,199)(150,200);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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*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(200)!(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)
( 13, 18)( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)
( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)
( 59, 74)( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)
( 82, 97)( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)
( 90, 95)(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)
(113,118)(114,119)(115,120)(131,146)(132,147)(133,148)(134,149)(135,150)
(136,141)(137,142)(138,143)(139,144)(140,145)(156,171)(157,172)(158,173)
(159,174)(160,175)(161,166)(162,167)(163,168)(164,169)(165,170)(181,196)
(182,197)(183,198)(184,199)(185,200)(186,191)(187,192)(188,193)(189,194)
(190,195);
s1 := 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)(101,131)(102,135)(103,134)(104,133)(105,132)(106,126)(107,130)
(108,129)(109,128)(110,127)(111,146)(112,150)(113,149)(114,148)(115,147)
(116,141)(117,145)(118,144)(119,143)(120,142)(121,136)(122,140)(123,139)
(124,138)(125,137)(151,181)(152,185)(153,184)(154,183)(155,182)(156,176)
(157,180)(158,179)(159,178)(160,177)(161,196)(162,200)(163,199)(164,198)
(165,197)(166,191)(167,195)(168,194)(169,193)(170,192)(171,186)(172,190)
(173,189)(174,188)(175,187);
s2 := Sym(200)!(  1,102)(  2,101)(  3,105)(  4,104)(  5,103)(  6,107)(  7,106)
(  8,110)(  9,109)( 10,108)( 11,112)( 12,111)( 13,115)( 14,114)( 15,113)
( 16,117)( 17,116)( 18,120)( 19,119)( 20,118)( 21,122)( 22,121)( 23,125)
( 24,124)( 25,123)( 26,127)( 27,126)( 28,130)( 29,129)( 30,128)( 31,132)
( 32,131)( 33,135)( 34,134)( 35,133)( 36,137)( 37,136)( 38,140)( 39,139)
( 40,138)( 41,142)( 42,141)( 43,145)( 44,144)( 45,143)( 46,147)( 47,146)
( 48,150)( 49,149)( 50,148)( 51,152)( 52,151)( 53,155)( 54,154)( 55,153)
( 56,157)( 57,156)( 58,160)( 59,159)( 60,158)( 61,162)( 62,161)( 63,165)
( 64,164)( 65,163)( 66,167)( 67,166)( 68,170)( 69,169)( 70,168)( 71,172)
( 72,171)( 73,175)( 74,174)( 75,173)( 76,177)( 77,176)( 78,180)( 79,179)
( 80,178)( 81,182)( 82,181)( 83,185)( 84,184)( 85,183)( 86,187)( 87,186)
( 88,190)( 89,189)( 90,188)( 91,192)( 92,191)( 93,195)( 94,194)( 95,193)
( 96,197)( 97,196)( 98,200)( 99,199)(100,198);
s3 := Sym(200)!(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)
(108,158)(109,159)(110,160)(111,161)(112,162)(113,163)(114,164)(115,165)
(116,166)(117,167)(118,168)(119,169)(120,170)(121,171)(122,172)(123,173)
(124,174)(125,175)(126,176)(127,177)(128,178)(129,179)(130,180)(131,181)
(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)
(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,196)(147,197)
(148,198)(149,199)(150,200);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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*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.
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