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# Polytope of Type {4,20}

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