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Polytope of Type {40,10}

Atlas Canonical Name : {40,10}*800a
Also Known As : {40,10|2}. if this polytope has another name.
Group : SmallGroup(800,570)
Rank : 3
Schlafli Type : {40,10}
Number of vertices, edges, etc : 40, 200, 10
Order of s0s1s2 : 40
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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}*400a
4-fold quotients : {10,10}*200a
5-fold quotients : {40,2}*160, {8,10}*160
10-fold quotients : {20,2}*80, {4,10}*80
20-fold quotients : {2,10}*40, {10,2}*40
25-fold quotients : {8,2}*32
40-fold quotients : {2,5}*20, {5,2}*20
50-fold quotients : {4,2}*16
100-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {80,10}*1600a, {40,20}*1600c
Permutation Representation (GAP) :
```s0 := (  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)( 13, 18)
( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 41)
( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 76)( 52, 77)( 53, 78)( 54, 79)
( 55, 80)( 56, 96)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 61, 91)( 62, 92)
( 63, 93)( 64, 94)( 65, 95)( 66, 86)( 67, 87)( 68, 88)( 69, 89)( 70, 90)
( 71, 81)( 72, 82)( 73, 83)( 74, 84)( 75, 85)(101,151)(102,152)(103,153)
(104,154)(105,155)(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)
(112,167)(113,168)(114,169)(115,170)(116,161)(117,162)(118,163)(119,164)
(120,165)(121,156)(122,157)(123,158)(124,159)(125,160)(126,176)(127,177)
(128,178)(129,179)(130,180)(131,196)(132,197)(133,198)(134,199)(135,200)
(136,191)(137,192)(138,193)(139,194)(140,195)(141,186)(142,187)(143,188)
(144,189)(145,190)(146,181)(147,182)(148,183)(149,184)(150,185);;
s1 := (  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)( 51,181)( 52,185)( 53,184)( 54,183)( 55,182)( 56,176)
( 57,180)( 58,179)( 59,178)( 60,177)( 61,196)( 62,200)( 63,199)( 64,198)
( 65,197)( 66,191)( 67,195)( 68,194)( 69,193)( 70,192)( 71,186)( 72,190)
( 73,189)( 74,188)( 75,187)( 76,156)( 77,160)( 78,159)( 79,158)( 80,157)
( 81,151)( 82,155)( 83,154)( 84,153)( 85,152)( 86,171)( 87,175)( 88,174)
( 89,173)( 90,172)( 91,166)( 92,170)( 93,169)( 94,168)( 95,167)( 96,161)
( 97,165)( 98,164)( 99,163)(100,162);;
s2 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)
(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)(118,120)
(121,122)(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)(138,140)
(141,142)(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)(158,160)
(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)(178,180)
(181,182)(183,185)(186,187)(188,190)(191,192)(193,195)(196,197)(198,200);;
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, 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,
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*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)!(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)
( 13, 18)( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)
( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 76)( 52, 77)( 53, 78)
( 54, 79)( 55, 80)( 56, 96)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 61, 91)
( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 86)( 67, 87)( 68, 88)( 69, 89)
( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)( 75, 85)(101,151)(102,152)
(103,153)(104,154)(105,155)(106,171)(107,172)(108,173)(109,174)(110,175)
(111,166)(112,167)(113,168)(114,169)(115,170)(116,161)(117,162)(118,163)
(119,164)(120,165)(121,156)(122,157)(123,158)(124,159)(125,160)(126,176)
(127,177)(128,178)(129,179)(130,180)(131,196)(132,197)(133,198)(134,199)
(135,200)(136,191)(137,192)(138,193)(139,194)(140,195)(141,186)(142,187)
(143,188)(144,189)(145,190)(146,181)(147,182)(148,183)(149,184)(150,185);
s1 := 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)( 51,181)( 52,185)( 53,184)( 54,183)( 55,182)
( 56,176)( 57,180)( 58,179)( 59,178)( 60,177)( 61,196)( 62,200)( 63,199)
( 64,198)( 65,197)( 66,191)( 67,195)( 68,194)( 69,193)( 70,192)( 71,186)
( 72,190)( 73,189)( 74,188)( 75,187)( 76,156)( 77,160)( 78,159)( 79,158)
( 80,157)( 81,151)( 82,155)( 83,154)( 84,153)( 85,152)( 86,171)( 87,175)
( 88,174)( 89,173)( 90,172)( 91,166)( 92,170)( 93,169)( 94,168)( 95,167)
( 96,161)( 97,165)( 98,164)( 99,163)(100,162);
s2 := Sym(200)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)
( 98,100)(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)
(118,120)(121,122)(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)
(138,140)(141,142)(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)
(158,160)(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)
(178,180)(181,182)(183,185)(186,187)(188,190)(191,192)(193,195)(196,197)
(198,200);
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, 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,
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*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