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# Polytope of Type {4,10,10}

Atlas Canonical Name : {4,10,10}*800c
if this polytope has a name.
Group : SmallGroup(800,1153)
Rank : 4
Schlafli Type : {4,10,10}
Number of vertices, edges, etc : 4, 20, 50, 10
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,10,10,2} of size 1600
Vertex Figure Of :
{2,4,10,10} of size 1600
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,10,5}*400, {2,10,10}*400b
4-fold quotients : {2,10,5}*200
5-fold quotients : {4,2,10}*160
10-fold quotients : {4,2,5}*80, {2,2,10}*80
20-fold quotients : {2,2,5}*40
25-fold quotients : {4,2,2}*32
50-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,10,20}*1600b, {8,10,10}*1600c, {4,20,10}*1600c
Permutation Representation (GAP) :
```s0 := (  1, 51)(  2, 52)(  3, 53)(  4, 54)(  5, 55)(  6, 56)(  7, 57)(  8, 58)
(  9, 59)( 10, 60)( 11, 61)( 12, 62)( 13, 63)( 14, 64)( 15, 65)( 16, 66)
( 17, 67)( 18, 68)( 19, 69)( 20, 70)( 21, 71)( 22, 72)( 23, 73)( 24, 74)
( 25, 75)( 26, 76)( 27, 77)( 28, 78)( 29, 79)( 30, 80)( 31, 81)( 32, 82)
( 33, 83)( 34, 84)( 35, 85)( 36, 86)( 37, 87)( 38, 88)( 39, 89)( 40, 90)
( 41, 91)( 42, 92)( 43, 93)( 44, 94)( 45, 95)( 46, 96)( 47, 97)( 48, 98)
( 49, 99)( 50,100)(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);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 51, 76)( 52, 80)( 53, 79)( 54, 78)
( 55, 77)( 56, 81)( 57, 85)( 58, 84)( 59, 83)( 60, 82)( 61, 86)( 62, 90)
( 63, 89)( 64, 88)( 65, 87)( 66, 91)( 67, 95)( 68, 94)( 69, 93)( 70, 92)
( 71, 96)( 72,100)( 73, 99)( 74, 98)( 75, 97)(102,105)(103,104)(107,110)
(108,109)(112,115)(113,114)(117,120)(118,119)(122,125)(123,124)(127,130)
(128,129)(132,135)(133,134)(137,140)(138,139)(142,145)(143,144)(147,150)
(148,149)(151,176)(152,180)(153,179)(154,178)(155,177)(156,181)(157,185)
(158,184)(159,183)(160,182)(161,186)(162,190)(163,189)(164,188)(165,187)
(166,191)(167,195)(168,194)(169,193)(170,192)(171,196)(172,200)(173,199)
(174,198)(175,197);;
s2 := (  1,  2)(  3,  5)(  6, 22)(  7, 21)(  8, 25)(  9, 24)( 10, 23)( 11, 17)
( 12, 16)( 13, 20)( 14, 19)( 15, 18)( 26, 27)( 28, 30)( 31, 47)( 32, 46)
( 33, 50)( 34, 49)( 35, 48)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)
( 51, 52)( 53, 55)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)( 61, 67)
( 62, 66)( 63, 70)( 64, 69)( 65, 68)( 76, 77)( 78, 80)( 81, 97)( 82, 96)
( 83,100)( 84, 99)( 85, 98)( 86, 92)( 87, 91)( 88, 95)( 89, 94)( 90, 93)
(101,102)(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)(111,117)
(112,116)(113,120)(114,119)(115,118)(126,127)(128,130)(131,147)(132,146)
(133,150)(134,149)(135,148)(136,142)(137,141)(138,145)(139,144)(140,143)
(151,152)(153,155)(156,172)(157,171)(158,175)(159,174)(160,173)(161,167)
(162,166)(163,170)(164,169)(165,168)(176,177)(178,180)(181,197)(182,196)
(183,200)(184,199)(185,198)(186,192)(187,191)(188,195)(189,194)(190,193);;
s3 := (  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,101)(  7,105)(  8,104)
(  9,103)( 10,102)( 11,121)( 12,125)( 13,124)( 14,123)( 15,122)( 16,116)
( 17,120)( 18,119)( 19,118)( 20,117)( 21,111)( 22,115)( 23,114)( 24,113)
( 25,112)( 26,131)( 27,135)( 28,134)( 29,133)( 30,132)( 31,126)( 32,130)
( 33,129)( 34,128)( 35,127)( 36,146)( 37,150)( 38,149)( 39,148)( 40,147)
( 41,141)( 42,145)( 43,144)( 44,143)( 45,142)( 46,136)( 47,140)( 48,139)
( 49,138)( 50,137)( 51,156)( 52,160)( 53,159)( 54,158)( 55,157)( 56,151)
( 57,155)( 58,154)( 59,153)( 60,152)( 61,171)( 62,175)( 63,174)( 64,173)
( 65,172)( 66,166)( 67,170)( 68,169)( 69,168)( 70,167)( 71,161)( 72,165)
( 73,164)( 74,163)( 75,162)( 76,181)( 77,185)( 78,184)( 79,183)( 80,182)
( 81,176)( 82,180)( 83,179)( 84,178)( 85,177)( 86,196)( 87,200)( 88,199)
( 89,198)( 90,197)( 91,191)( 92,195)( 93,194)( 94,193)( 95,192)( 96,186)
( 97,190)( 98,189)( 99,188)(100,187);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(200)!(  1, 51)(  2, 52)(  3, 53)(  4, 54)(  5, 55)(  6, 56)(  7, 57)
(  8, 58)(  9, 59)( 10, 60)( 11, 61)( 12, 62)( 13, 63)( 14, 64)( 15, 65)
( 16, 66)( 17, 67)( 18, 68)( 19, 69)( 20, 70)( 21, 71)( 22, 72)( 23, 73)
( 24, 74)( 25, 75)( 26, 76)( 27, 77)( 28, 78)( 29, 79)( 30, 80)( 31, 81)
( 32, 82)( 33, 83)( 34, 84)( 35, 85)( 36, 86)( 37, 87)( 38, 88)( 39, 89)
( 40, 90)( 41, 91)( 42, 92)( 43, 93)( 44, 94)( 45, 95)( 46, 96)( 47, 97)
( 48, 98)( 49, 99)( 50,100)(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);
s1 := Sym(200)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 51, 76)( 52, 80)( 53, 79)
( 54, 78)( 55, 77)( 56, 81)( 57, 85)( 58, 84)( 59, 83)( 60, 82)( 61, 86)
( 62, 90)( 63, 89)( 64, 88)( 65, 87)( 66, 91)( 67, 95)( 68, 94)( 69, 93)
( 70, 92)( 71, 96)( 72,100)( 73, 99)( 74, 98)( 75, 97)(102,105)(103,104)
(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)(122,125)(123,124)
(127,130)(128,129)(132,135)(133,134)(137,140)(138,139)(142,145)(143,144)
(147,150)(148,149)(151,176)(152,180)(153,179)(154,178)(155,177)(156,181)
(157,185)(158,184)(159,183)(160,182)(161,186)(162,190)(163,189)(164,188)
(165,187)(166,191)(167,195)(168,194)(169,193)(170,192)(171,196)(172,200)
(173,199)(174,198)(175,197);
s2 := Sym(200)!(  1,  2)(  3,  5)(  6, 22)(  7, 21)(  8, 25)(  9, 24)( 10, 23)
( 11, 17)( 12, 16)( 13, 20)( 14, 19)( 15, 18)( 26, 27)( 28, 30)( 31, 47)
( 32, 46)( 33, 50)( 34, 49)( 35, 48)( 36, 42)( 37, 41)( 38, 45)( 39, 44)
( 40, 43)( 51, 52)( 53, 55)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)
( 61, 67)( 62, 66)( 63, 70)( 64, 69)( 65, 68)( 76, 77)( 78, 80)( 81, 97)
( 82, 96)( 83,100)( 84, 99)( 85, 98)( 86, 92)( 87, 91)( 88, 95)( 89, 94)
( 90, 93)(101,102)(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)
(111,117)(112,116)(113,120)(114,119)(115,118)(126,127)(128,130)(131,147)
(132,146)(133,150)(134,149)(135,148)(136,142)(137,141)(138,145)(139,144)
(140,143)(151,152)(153,155)(156,172)(157,171)(158,175)(159,174)(160,173)
(161,167)(162,166)(163,170)(164,169)(165,168)(176,177)(178,180)(181,197)
(182,196)(183,200)(184,199)(185,198)(186,192)(187,191)(188,195)(189,194)
(190,193);
s3 := Sym(200)!(  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,101)(  7,105)
(  8,104)(  9,103)( 10,102)( 11,121)( 12,125)( 13,124)( 14,123)( 15,122)
( 16,116)( 17,120)( 18,119)( 19,118)( 20,117)( 21,111)( 22,115)( 23,114)
( 24,113)( 25,112)( 26,131)( 27,135)( 28,134)( 29,133)( 30,132)( 31,126)
( 32,130)( 33,129)( 34,128)( 35,127)( 36,146)( 37,150)( 38,149)( 39,148)
( 40,147)( 41,141)( 42,145)( 43,144)( 44,143)( 45,142)( 46,136)( 47,140)
( 48,139)( 49,138)( 50,137)( 51,156)( 52,160)( 53,159)( 54,158)( 55,157)
( 56,151)( 57,155)( 58,154)( 59,153)( 60,152)( 61,171)( 62,175)( 63,174)
( 64,173)( 65,172)( 66,166)( 67,170)( 68,169)( 69,168)( 70,167)( 71,161)
( 72,165)( 73,164)( 74,163)( 75,162)( 76,181)( 77,185)( 78,184)( 79,183)
( 80,182)( 81,176)( 82,180)( 83,179)( 84,178)( 85,177)( 86,196)( 87,200)
( 88,199)( 89,198)( 90,197)( 91,191)( 92,195)( 93,194)( 94,193)( 95,192)
( 96,186)( 97,190)( 98,189)( 99,188)(100,187);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope