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# Polytope of Type {4,100}

Atlas Canonical Name : {4,100}*800
Also Known As : {4,100|2}. if this polytope has another name.
Group : SmallGroup(800,103)
Rank : 3
Schlafli Type : {4,100}
Number of vertices, edges, etc : 4, 200, 100
Order of s0s1s2 : 100
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,100,2} of size 1600
Vertex Figure Of :
{2,4,100} of size 1600
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,100}*400, {4,50}*400
4-fold quotients : {2,50}*200
5-fold quotients : {4,20}*160
8-fold quotients : {2,25}*100
10-fold quotients : {2,20}*80, {4,10}*80
20-fold quotients : {2,10}*40
25-fold quotients : {4,4}*32
40-fold quotients : {2,5}*20
50-fold quotients : {2,4}*16, {4,2}*16
100-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,200}*1600a, {4,100}*1600, {4,200}*1600b, {8,100}*1600a, {8,100}*1600b
Permutation Representation (GAP) :
```s0 := (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);;
s1 := (  1,101)(  2,105)(  3,104)(  4,103)(  5,102)(  6,125)(  7,124)(  8,123)
(  9,122)( 10,121)( 11,120)( 12,119)( 13,118)( 14,117)( 15,116)( 16,115)
( 17,114)( 18,113)( 19,112)( 20,111)( 21,110)( 22,109)( 23,108)( 24,107)
( 25,106)( 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)( 51,151)( 52,155)( 53,154)( 54,153)( 55,152)( 56,175)
( 57,174)( 58,173)( 59,172)( 60,171)( 61,170)( 62,169)( 63,168)( 64,167)
( 65,166)( 66,165)( 67,164)( 68,163)( 69,162)( 70,161)( 71,160)( 72,159)
( 73,158)( 74,157)( 75,156)( 76,176)( 77,180)( 78,179)( 79,178)( 80,177)
( 81,200)( 82,199)( 83,198)( 84,197)( 85,196)( 86,195)( 87,194)( 88,193)
( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,187)( 95,186)( 96,185)
( 97,184)( 98,183)( 99,182)(100,181);;
s2 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 25)( 12, 24)( 13, 23)
( 14, 22)( 15, 21)( 16, 20)( 17, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)
( 30, 32)( 36, 50)( 37, 49)( 38, 48)( 39, 47)( 40, 46)( 41, 45)( 42, 44)
( 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, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 86,100)( 87, 99)( 88, 98)( 89, 97)( 90, 96)( 91, 95)( 92, 94)
(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)(126,181)(127,185)(128,184)(129,183)(130,182)(131,176)(132,180)
(133,179)(134,178)(135,177)(136,200)(137,199)(138,198)(139,197)(140,196)
(141,195)(142,194)(143,193)(144,192)(145,191)(146,190)(147,189)(148,188)
(149,187)(150,186);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(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);
s1 := Sym(200)!(  1,101)(  2,105)(  3,104)(  4,103)(  5,102)(  6,125)(  7,124)
(  8,123)(  9,122)( 10,121)( 11,120)( 12,119)( 13,118)( 14,117)( 15,116)
( 16,115)( 17,114)( 18,113)( 19,112)( 20,111)( 21,110)( 22,109)( 23,108)
( 24,107)( 25,106)( 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)( 51,151)( 52,155)( 53,154)( 54,153)( 55,152)
( 56,175)( 57,174)( 58,173)( 59,172)( 60,171)( 61,170)( 62,169)( 63,168)
( 64,167)( 65,166)( 66,165)( 67,164)( 68,163)( 69,162)( 70,161)( 71,160)
( 72,159)( 73,158)( 74,157)( 75,156)( 76,176)( 77,180)( 78,179)( 79,178)
( 80,177)( 81,200)( 82,199)( 83,198)( 84,197)( 85,196)( 86,195)( 87,194)
( 88,193)( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,187)( 95,186)
( 96,185)( 97,184)( 98,183)( 99,182)(100,181);
s2 := Sym(200)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 25)( 12, 24)
( 13, 23)( 14, 22)( 15, 21)( 16, 20)( 17, 19)( 26, 31)( 27, 35)( 28, 34)
( 29, 33)( 30, 32)( 36, 50)( 37, 49)( 38, 48)( 39, 47)( 40, 46)( 41, 45)
( 42, 44)( 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, 81)( 77, 85)( 78, 84)
( 79, 83)( 80, 82)( 86,100)( 87, 99)( 88, 98)( 89, 97)( 90, 96)( 91, 95)
( 92, 94)(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)(126,181)(127,185)(128,184)(129,183)(130,182)(131,176)
(132,180)(133,179)(134,178)(135,177)(136,200)(137,199)(138,198)(139,197)
(140,196)(141,195)(142,194)(143,193)(144,192)(145,191)(146,190)(147,189)
(148,188)(149,187)(150,186);
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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