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# Polytope of Type {20,4}

Atlas Canonical Name : {20,4}*1600
Also Known As : {20,4}4if this polytope has another name.
Group : SmallGroup(1600,6624)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 200, 400, 40
Order of s0s1s2 : 4
Order of s0s1s2s1 : 20
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,4}*800
4-fold quotients : {10,4}*400
8-fold quotients : {10,4}*200
25-fold quotients : {4,4}*64
50-fold quotients : {4,4}*32
100-fold quotients : {2,4}*16, {4,2}*16
200-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,126)(  2,130)(  3,129)(  4,128)(  5,127)(  6,146)(  7,150)(  8,149)
(  9,148)( 10,147)( 11,141)( 12,145)( 13,144)( 14,143)( 15,142)( 16,136)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,131)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,121)( 32,125)
( 33,124)( 34,123)( 35,122)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)
( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46,106)( 47,110)( 48,109)
( 49,108)( 50,107)( 51,176)( 52,180)( 53,179)( 54,178)( 55,177)( 56,196)
( 57,200)( 58,199)( 59,198)( 60,197)( 61,191)( 62,195)( 63,194)( 64,193)
( 65,192)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,181)( 72,185)
( 73,184)( 74,183)( 75,182)( 76,151)( 77,155)( 78,154)( 79,153)( 80,152)
( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,166)( 87,170)( 88,169)
( 89,168)( 90,167)( 91,161)( 92,165)( 93,164)( 94,163)( 95,162)( 96,156)
( 97,160)( 98,159)( 99,158)(100,157);;
s1 := (  1,  8)(  2, 11)(  3, 19)(  4, 22)(  6, 20)(  7, 23)( 10, 12)( 14, 16)
( 15, 24)( 18, 25)( 26, 33)( 27, 36)( 28, 44)( 29, 47)( 31, 45)( 32, 48)
( 35, 37)( 39, 41)( 40, 49)( 43, 50)( 51, 58)( 52, 61)( 53, 69)( 54, 72)
( 56, 70)( 57, 73)( 60, 62)( 64, 66)( 65, 74)( 68, 75)( 76, 83)( 77, 86)
( 78, 94)( 79, 97)( 81, 95)( 82, 98)( 85, 87)( 89, 91)( 90, 99)( 93,100)
(101,158)(102,161)(103,169)(104,172)(105,155)(106,170)(107,173)(108,151)
(109,159)(110,162)(111,152)(112,160)(113,163)(114,166)(115,174)(116,164)
(117,167)(118,175)(119,153)(120,156)(121,171)(122,154)(123,157)(124,165)
(125,168)(126,183)(127,186)(128,194)(129,197)(130,180)(131,195)(132,198)
(133,176)(134,184)(135,187)(136,177)(137,185)(138,188)(139,191)(140,199)
(141,189)(142,192)(143,200)(144,178)(145,181)(146,196)(147,179)(148,182)
(149,190)(150,193);;
s2 := (  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)( 11, 16)
( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)( 33, 43)
( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 51, 76)( 52, 87)( 53, 98)( 54, 84)
( 55, 95)( 56, 96)( 57, 82)( 58, 93)( 59, 79)( 60, 90)( 61, 91)( 62, 77)
( 63, 88)( 64, 99)( 65, 85)( 66, 86)( 67, 97)( 68, 83)( 69, 94)( 70, 80)
( 71, 81)( 72, 92)( 73, 78)( 74, 89)( 75,100)(102,112)(103,123)(104,109)
(105,120)(106,121)(108,118)(110,115)(111,116)(114,124)(117,122)(127,137)
(128,148)(129,134)(130,145)(131,146)(133,143)(135,140)(136,141)(139,149)
(142,147)(151,176)(152,187)(153,198)(154,184)(155,195)(156,196)(157,182)
(158,193)(159,179)(160,190)(161,191)(162,177)(163,188)(164,199)(165,185)
(166,186)(167,197)(168,183)(169,194)(170,180)(171,181)(172,192)(173,178)
(174,189)(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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope