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# Polytope of Type {100,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {100,4}*800
Also Known As : {100,4|2}. if this polytope has another name.
Group : SmallGroup(800,103)
Rank : 3
Schlafli Type : {100,4}
Number of vertices, edges, etc : 100, 200, 4
Order of s0s1s2 : 100
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{100,4,2} of size 1600
Vertex Figure Of :
{2,100,4} of size 1600
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {100,2}*400, {50,4}*400
4-fold quotients : {50,2}*200
5-fold quotients : {20,4}*160
8-fold quotients : {25,2}*100
10-fold quotients : {20,2}*80, {10,4}*80
20-fold quotients : {10,2}*40
25-fold quotients : {4,4}*32
40-fold quotients : {5,2}*20
50-fold quotients : {2,4}*16, {4,2}*16
100-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {200,4}*1600a, {100,4}*1600, {200,4}*1600b, {100,8}*1600a, {100,8}*1600b
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  6, 25)(  7, 24)(  8, 23)(  9, 22)( 10, 21)( 11, 20)
( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 27, 30)( 28, 29)( 31, 50)( 32, 49)
( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 37, 44)( 38, 43)( 39, 42)( 40, 41)
( 52, 55)( 53, 54)( 56, 75)( 57, 74)( 58, 73)( 59, 72)( 60, 71)( 61, 70)
( 62, 69)( 63, 68)( 64, 67)( 65, 66)( 77, 80)( 78, 79)( 81,100)( 82, 99)
( 83, 98)( 84, 97)( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 89, 92)( 90, 91)
(101,151)(102,155)(103,154)(104,153)(105,152)(106,175)(107,174)(108,173)
(109,172)(110,171)(111,170)(112,169)(113,168)(114,167)(115,166)(116,165)
(117,164)(118,163)(119,162)(120,161)(121,160)(122,159)(123,158)(124,157)
(125,156)(126,176)(127,180)(128,179)(129,178)(130,177)(131,200)(132,199)
(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)
(141,190)(142,189)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)
(149,182)(150,181);;
s1 := (  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,101)(  7,105)(  8,104)
(  9,103)( 10,102)( 11,125)( 12,124)( 13,123)( 14,122)( 15,121)( 16,120)
( 17,119)( 18,118)( 19,117)( 20,116)( 21,115)( 22,114)( 23,113)( 24,112)
( 25,111)( 26,131)( 27,135)( 28,134)( 29,133)( 30,132)( 31,126)( 32,130)
( 33,129)( 34,128)( 35,127)( 36,150)( 37,149)( 38,148)( 39,147)( 40,146)
( 41,145)( 42,144)( 43,143)( 44,142)( 45,141)( 46,140)( 47,139)( 48,138)
( 49,137)( 50,136)( 51,156)( 52,160)( 53,159)( 54,158)( 55,157)( 56,151)
( 57,155)( 58,154)( 59,153)( 60,152)( 61,175)( 62,174)( 63,173)( 64,172)
( 65,171)( 66,170)( 67,169)( 68,168)( 69,167)( 70,166)( 71,165)( 72,164)
( 73,163)( 74,162)( 75,161)( 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);;
s2 := (101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)
(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)
(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)
(125,150)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181)(157,182)
(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)
(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)
(174,199)(175,200);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!(  2,  5)(  3,  4)(  6, 25)(  7, 24)(  8, 23)(  9, 22)( 10, 21)
( 11, 20)( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 27, 30)( 28, 29)( 31, 50)
( 32, 49)( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 37, 44)( 38, 43)( 39, 42)
( 40, 41)( 52, 55)( 53, 54)( 56, 75)( 57, 74)( 58, 73)( 59, 72)( 60, 71)
( 61, 70)( 62, 69)( 63, 68)( 64, 67)( 65, 66)( 77, 80)( 78, 79)( 81,100)
( 82, 99)( 83, 98)( 84, 97)( 85, 96)( 86, 95)( 87, 94)( 88, 93)( 89, 92)
( 90, 91)(101,151)(102,155)(103,154)(104,153)(105,152)(106,175)(107,174)
(108,173)(109,172)(110,171)(111,170)(112,169)(113,168)(114,167)(115,166)
(116,165)(117,164)(118,163)(119,162)(120,161)(121,160)(122,159)(123,158)
(124,157)(125,156)(126,176)(127,180)(128,179)(129,178)(130,177)(131,200)
(132,199)(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,192)
(140,191)(141,190)(142,189)(143,188)(144,187)(145,186)(146,185)(147,184)
(148,183)(149,182)(150,181);
s1 := Sym(200)!(  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,101)(  7,105)
(  8,104)(  9,103)( 10,102)( 11,125)( 12,124)( 13,123)( 14,122)( 15,121)
( 16,120)( 17,119)( 18,118)( 19,117)( 20,116)( 21,115)( 22,114)( 23,113)
( 24,112)( 25,111)( 26,131)( 27,135)( 28,134)( 29,133)( 30,132)( 31,126)
( 32,130)( 33,129)( 34,128)( 35,127)( 36,150)( 37,149)( 38,148)( 39,147)
( 40,146)( 41,145)( 42,144)( 43,143)( 44,142)( 45,141)( 46,140)( 47,139)
( 48,138)( 49,137)( 50,136)( 51,156)( 52,160)( 53,159)( 54,158)( 55,157)
( 56,151)( 57,155)( 58,154)( 59,153)( 60,152)( 61,175)( 62,174)( 63,173)
( 64,172)( 65,171)( 66,170)( 67,169)( 68,168)( 69,167)( 70,166)( 71,165)
( 72,164)( 73,163)( 74,162)( 75,161)( 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);
s2 := Sym(200)!(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)
(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)
(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)
(124,149)(125,150)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181)
(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)
(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)
(173,198)(174,199)(175,200);
poly := sub<Sym(200)|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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope