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# Polytope of Type {40,10}

Atlas Canonical Name : {40,10}*800c
if this polytope has a name.
Group : SmallGroup(800,703)
Rank : 3
Schlafli Type : {40,10}
Number of vertices, edges, etc : 40, 200, 10
Order of s0s1s2 : 40
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{40,10,2} of size 1600
Vertex Figure Of :
{2,40,10} of size 1600
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,10}*400c
4-fold quotients : {10,10}*200b
5-fold quotients : {8,10}*160
8-fold quotients : {10,5}*100
10-fold quotients : {4,10}*80
20-fold quotients : {2,10}*40
25-fold quotients : {8,2}*32
40-fold quotients : {2,5}*20
50-fold quotients : {4,2}*16
100-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {40,20}*1600b, {80,10}*1600c
Permutation Representation (GAP) :
```s0 := (  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)(101,151)(102,155)(103,154)
(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,161)
(112,165)(113,164)(114,163)(115,162)(116,166)(117,170)(118,169)(119,168)
(120,167)(121,171)(122,175)(123,174)(124,173)(125,172)(126,176)(127,180)
(128,179)(129,178)(130,177)(131,181)(132,185)(133,184)(134,183)(135,182)
(136,186)(137,190)(138,189)(139,188)(140,187)(141,191)(142,195)(143,194)
(144,193)(145,192)(146,196)(147,200)(148,199)(149,198)(150,197);;
s1 := (  1,102)(  2,101)(  3,105)(  4,104)(  5,103)(  6,122)(  7,121)(  8,125)
(  9,124)( 10,123)( 11,117)( 12,116)( 13,120)( 14,119)( 15,118)( 16,112)
( 17,111)( 18,115)( 19,114)( 20,113)( 21,107)( 22,106)( 23,110)( 24,109)
( 25,108)( 26,127)( 27,126)( 28,130)( 29,129)( 30,128)( 31,147)( 32,146)
( 33,150)( 34,149)( 35,148)( 36,142)( 37,141)( 38,145)( 39,144)( 40,143)
( 41,137)( 42,136)( 43,140)( 44,139)( 45,138)( 46,132)( 47,131)( 48,135)
( 49,134)( 50,133)( 51,177)( 52,176)( 53,180)( 54,179)( 55,178)( 56,197)
( 57,196)( 58,200)( 59,199)( 60,198)( 61,192)( 62,191)( 63,195)( 64,194)
( 65,193)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,182)( 72,181)
( 73,185)( 74,184)( 75,183)( 76,152)( 77,151)( 78,155)( 79,154)( 80,153)
( 81,172)( 82,171)( 83,175)( 84,174)( 85,173)( 86,167)( 87,166)( 88,170)
( 89,169)( 90,168)( 91,162)( 92,161)( 93,165)( 94,164)( 95,163)( 96,157)
( 97,156)( 98,160)( 99,159)(100,158);;
s2 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 21)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)
( 30, 32)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 42, 45)( 43, 44)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 71)( 62, 75)( 63, 74)
( 64, 73)( 65, 72)( 67, 70)( 68, 69)( 76, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)( 93, 94)
(101,106)(102,110)(103,109)(104,108)(105,107)(111,121)(112,125)(113,124)
(114,123)(115,122)(117,120)(118,119)(126,131)(127,135)(128,134)(129,133)
(130,132)(136,146)(137,150)(138,149)(139,148)(140,147)(142,145)(143,144)
(151,156)(152,160)(153,159)(154,158)(155,157)(161,171)(162,175)(163,174)
(164,173)(165,172)(167,170)(168,169)(176,181)(177,185)(178,184)(179,183)
(180,182)(186,196)(187,200)(188,199)(189,198)(190,197)(192,195)(193,194);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
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,  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)(101,151)(102,155)
(103,154)(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)
(111,161)(112,165)(113,164)(114,163)(115,162)(116,166)(117,170)(118,169)
(119,168)(120,167)(121,171)(122,175)(123,174)(124,173)(125,172)(126,176)
(127,180)(128,179)(129,178)(130,177)(131,181)(132,185)(133,184)(134,183)
(135,182)(136,186)(137,190)(138,189)(139,188)(140,187)(141,191)(142,195)
(143,194)(144,193)(145,192)(146,196)(147,200)(148,199)(149,198)(150,197);
s1 := Sym(200)!(  1,102)(  2,101)(  3,105)(  4,104)(  5,103)(  6,122)(  7,121)
(  8,125)(  9,124)( 10,123)( 11,117)( 12,116)( 13,120)( 14,119)( 15,118)
( 16,112)( 17,111)( 18,115)( 19,114)( 20,113)( 21,107)( 22,106)( 23,110)
( 24,109)( 25,108)( 26,127)( 27,126)( 28,130)( 29,129)( 30,128)( 31,147)
( 32,146)( 33,150)( 34,149)( 35,148)( 36,142)( 37,141)( 38,145)( 39,144)
( 40,143)( 41,137)( 42,136)( 43,140)( 44,139)( 45,138)( 46,132)( 47,131)
( 48,135)( 49,134)( 50,133)( 51,177)( 52,176)( 53,180)( 54,179)( 55,178)
( 56,197)( 57,196)( 58,200)( 59,199)( 60,198)( 61,192)( 62,191)( 63,195)
( 64,194)( 65,193)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,182)
( 72,181)( 73,185)( 74,184)( 75,183)( 76,152)( 77,151)( 78,155)( 79,154)
( 80,153)( 81,172)( 82,171)( 83,175)( 84,174)( 85,173)( 86,167)( 87,166)
( 88,170)( 89,169)( 90,168)( 91,162)( 92,161)( 93,165)( 94,164)( 95,163)
( 96,157)( 97,156)( 98,160)( 99,159)(100,158);
s2 := Sym(200)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 21)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)
( 29, 33)( 30, 32)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 42, 45)
( 43, 44)( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 71)( 62, 75)
( 63, 74)( 64, 73)( 65, 72)( 67, 70)( 68, 69)( 76, 81)( 77, 85)( 78, 84)
( 79, 83)( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)
( 93, 94)(101,106)(102,110)(103,109)(104,108)(105,107)(111,121)(112,125)
(113,124)(114,123)(115,122)(117,120)(118,119)(126,131)(127,135)(128,134)
(129,133)(130,132)(136,146)(137,150)(138,149)(139,148)(140,147)(142,145)
(143,144)(151,156)(152,160)(153,159)(154,158)(155,157)(161,171)(162,175)
(163,174)(164,173)(165,172)(167,170)(168,169)(176,181)(177,185)(178,184)
(179,183)(180,182)(186,196)(187,200)(188,199)(189,198)(190,197)(192,195)
(193,194);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
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