# Polytope of Type {10,12}

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

```
References : None.
to this polytope