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# Polytope of Type {10,15}

Atlas Canonical Name : {10,15}*960
if this polytope has a name.
Group : SmallGroup(960,11364)
Rank : 3
Schlafli Type : {10,15}
Number of vertices, edges, etc : 32, 240, 48
Order of s0s1s2 : 12
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,15,2} of size 1920
Vertex Figure Of :
{2,10,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {10,5}*320b
6-fold quotients : {5,5}*160
80-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,15}*1920a, {20,15}*1920b, {10,15}*1920, {10,30}*1920a, {10,30}*1920b
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)( 18, 75)
( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)( 26, 71)
( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)( 34, 59)
( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)( 42, 55)
( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 82, 91)( 83, 94)
( 84, 88)( 86, 95)( 87, 90)( 89, 93)( 97,145)( 98,155)( 99,158)(100,152)
(101,149)(102,159)(103,154)(104,148)(105,157)(106,151)(107,146)(108,156)
(109,153)(110,147)(111,150)(112,160)(113,129)(114,139)(115,142)(116,136)
(117,133)(118,143)(119,138)(120,132)(121,141)(122,135)(123,130)(124,140)
(125,137)(126,131)(127,134)(128,144)(162,171)(163,174)(164,168)(166,175)
(167,170)(169,173)(177,225)(178,235)(179,238)(180,232)(181,229)(182,239)
(183,234)(184,228)(185,237)(186,231)(187,226)(188,236)(189,233)(190,227)
(191,230)(192,240)(193,209)(194,219)(195,222)(196,216)(197,213)(198,223)
(199,218)(200,212)(201,221)(202,215)(203,210)(204,220)(205,217)(206,211)
(207,214)(208,224);;
s1 := (  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 22)(  6, 21)(  7, 23)(  8, 24)
(  9, 32)( 10, 31)( 11, 29)( 12, 30)( 13, 27)( 14, 28)( 15, 26)( 16, 25)
( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 71)( 40, 72)
( 41, 80)( 42, 79)( 43, 77)( 44, 78)( 45, 75)( 46, 76)( 47, 74)( 48, 73)
( 51, 52)( 53, 54)( 57, 64)( 58, 63)( 59, 61)( 60, 62)( 81,177)( 82,178)
( 83,180)( 84,179)( 85,182)( 86,181)( 87,183)( 88,184)( 89,192)( 90,191)
( 91,189)( 92,190)( 93,187)( 94,188)( 95,186)( 96,185)( 97,161)( 98,162)
( 99,164)(100,163)(101,166)(102,165)(103,167)(104,168)(105,176)(106,175)
(107,173)(108,174)(109,171)(110,172)(111,170)(112,169)(113,225)(114,226)
(115,228)(116,227)(117,230)(118,229)(119,231)(120,232)(121,240)(122,239)
(123,237)(124,238)(125,235)(126,236)(127,234)(128,233)(129,209)(130,210)
(131,212)(132,211)(133,214)(134,213)(135,215)(136,216)(137,224)(138,223)
(139,221)(140,222)(141,219)(142,220)(143,218)(144,217)(145,193)(146,194)
(147,196)(148,195)(149,198)(150,197)(151,199)(152,200)(153,208)(154,207)
(155,205)(156,206)(157,203)(158,204)(159,202)(160,201);;
s2 := (  1, 85)(  2, 95)(  3, 90)(  4, 84)(  5, 81)(  6, 91)(  7, 94)(  8, 88)
(  9, 89)( 10, 83)( 11, 86)( 12, 96)( 13, 93)( 14, 87)( 15, 82)( 16, 92)
( 17,149)( 18,159)( 19,154)( 20,148)( 21,145)( 22,155)( 23,158)( 24,152)
( 25,153)( 26,147)( 27,150)( 28,160)( 29,157)( 30,151)( 31,146)( 32,156)
( 33,133)( 34,143)( 35,138)( 36,132)( 37,129)( 38,139)( 39,142)( 40,136)
( 41,137)( 42,131)( 43,134)( 44,144)( 45,141)( 46,135)( 47,130)( 48,140)
( 49,117)( 50,127)( 51,122)( 52,116)( 53,113)( 54,123)( 55,126)( 56,120)
( 57,121)( 58,115)( 59,118)( 60,128)( 61,125)( 62,119)( 63,114)( 64,124)
( 65,101)( 66,111)( 67,106)( 68,100)( 69, 97)( 70,107)( 71,110)( 72,104)
( 73,105)( 74, 99)( 75,102)( 76,112)( 77,109)( 78,103)( 79, 98)( 80,108)
(161,165)(162,175)(163,170)(166,171)(167,174)(172,176)(177,229)(178,239)
(179,234)(180,228)(181,225)(182,235)(183,238)(184,232)(185,233)(186,227)
(187,230)(188,240)(189,237)(190,231)(191,226)(192,236)(193,213)(194,223)
(195,218)(196,212)(197,209)(198,219)(199,222)(200,216)(201,217)(202,211)
(203,214)(204,224)(205,221)(206,215)(207,210)(208,220);;
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*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*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*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(240)!(  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)
( 18, 75)( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)
( 26, 71)( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)
( 34, 59)( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)
( 42, 55)( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 82, 91)
( 83, 94)( 84, 88)( 86, 95)( 87, 90)( 89, 93)( 97,145)( 98,155)( 99,158)
(100,152)(101,149)(102,159)(103,154)(104,148)(105,157)(106,151)(107,146)
(108,156)(109,153)(110,147)(111,150)(112,160)(113,129)(114,139)(115,142)
(116,136)(117,133)(118,143)(119,138)(120,132)(121,141)(122,135)(123,130)
(124,140)(125,137)(126,131)(127,134)(128,144)(162,171)(163,174)(164,168)
(166,175)(167,170)(169,173)(177,225)(178,235)(179,238)(180,232)(181,229)
(182,239)(183,234)(184,228)(185,237)(186,231)(187,226)(188,236)(189,233)
(190,227)(191,230)(192,240)(193,209)(194,219)(195,222)(196,216)(197,213)
(198,223)(199,218)(200,212)(201,221)(202,215)(203,210)(204,220)(205,217)
(206,211)(207,214)(208,224);
s1 := Sym(240)!(  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 22)(  6, 21)(  7, 23)
(  8, 24)(  9, 32)( 10, 31)( 11, 29)( 12, 30)( 13, 27)( 14, 28)( 15, 26)
( 16, 25)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 71)
( 40, 72)( 41, 80)( 42, 79)( 43, 77)( 44, 78)( 45, 75)( 46, 76)( 47, 74)
( 48, 73)( 51, 52)( 53, 54)( 57, 64)( 58, 63)( 59, 61)( 60, 62)( 81,177)
( 82,178)( 83,180)( 84,179)( 85,182)( 86,181)( 87,183)( 88,184)( 89,192)
( 90,191)( 91,189)( 92,190)( 93,187)( 94,188)( 95,186)( 96,185)( 97,161)
( 98,162)( 99,164)(100,163)(101,166)(102,165)(103,167)(104,168)(105,176)
(106,175)(107,173)(108,174)(109,171)(110,172)(111,170)(112,169)(113,225)
(114,226)(115,228)(116,227)(117,230)(118,229)(119,231)(120,232)(121,240)
(122,239)(123,237)(124,238)(125,235)(126,236)(127,234)(128,233)(129,209)
(130,210)(131,212)(132,211)(133,214)(134,213)(135,215)(136,216)(137,224)
(138,223)(139,221)(140,222)(141,219)(142,220)(143,218)(144,217)(145,193)
(146,194)(147,196)(148,195)(149,198)(150,197)(151,199)(152,200)(153,208)
(154,207)(155,205)(156,206)(157,203)(158,204)(159,202)(160,201);
s2 := Sym(240)!(  1, 85)(  2, 95)(  3, 90)(  4, 84)(  5, 81)(  6, 91)(  7, 94)
(  8, 88)(  9, 89)( 10, 83)( 11, 86)( 12, 96)( 13, 93)( 14, 87)( 15, 82)
( 16, 92)( 17,149)( 18,159)( 19,154)( 20,148)( 21,145)( 22,155)( 23,158)
( 24,152)( 25,153)( 26,147)( 27,150)( 28,160)( 29,157)( 30,151)( 31,146)
( 32,156)( 33,133)( 34,143)( 35,138)( 36,132)( 37,129)( 38,139)( 39,142)
( 40,136)( 41,137)( 42,131)( 43,134)( 44,144)( 45,141)( 46,135)( 47,130)
( 48,140)( 49,117)( 50,127)( 51,122)( 52,116)( 53,113)( 54,123)( 55,126)
( 56,120)( 57,121)( 58,115)( 59,118)( 60,128)( 61,125)( 62,119)( 63,114)
( 64,124)( 65,101)( 66,111)( 67,106)( 68,100)( 69, 97)( 70,107)( 71,110)
( 72,104)( 73,105)( 74, 99)( 75,102)( 76,112)( 77,109)( 78,103)( 79, 98)
( 80,108)(161,165)(162,175)(163,170)(166,171)(167,174)(172,176)(177,229)
(178,239)(179,234)(180,228)(181,225)(182,235)(183,238)(184,232)(185,233)
(186,227)(187,230)(188,240)(189,237)(190,231)(191,226)(192,236)(193,213)
(194,223)(195,218)(196,212)(197,209)(198,219)(199,222)(200,216)(201,217)
(202,211)(203,214)(204,224)(205,221)(206,215)(207,210)(208,220);
poly := sub<Sym(240)|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*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*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*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 >;

```
References : None.
to this polytope