# Polytope of Type {40,10}

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

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

```
References : None.
to this polytope