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# Polytope of Type {20,12}

Atlas Canonical Name : {20,12}*960c
if this polytope has a name.
Group : SmallGroup(960,10994)
Rank : 3
Schlafli Type : {20,12}
Number of vertices, edges, etc : 40, 240, 24
Order of s0s1s2 : 30
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{20,12,2} of size 1920
Vertex Figure Of :
{2,20,12} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,6}*480c
4-fold quotients : {20,6}*240b
5-fold quotients : {4,12}*192c
8-fold quotients : {10,6}*120
10-fold quotients : {4,6}*96
20-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {10,2}*40
40-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {5,2}*20
80-fold quotients : {2,3}*12
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,12}*1920c, {40,12}*1920g, {40,12}*1920h, {20,24}*1920e, {20,24}*1920f
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)( 10, 16)
( 11, 13)( 12, 14)( 21, 23)( 22, 24)( 25, 39)( 26, 40)( 27, 37)( 28, 38)
( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 41, 43)( 42, 44)( 45, 59)( 46, 60)
( 47, 57)( 48, 58)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 61, 63)( 62, 64)
( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)( 72, 74)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89, 95)( 90, 96)
( 91, 93)( 92, 94)(101,103)(102,104)(105,119)(106,120)(107,117)(108,118)
(109,115)(110,116)(111,113)(112,114)(121,183)(122,184)(123,181)(124,182)
(125,199)(126,200)(127,197)(128,198)(129,195)(130,196)(131,193)(132,194)
(133,191)(134,192)(135,189)(136,190)(137,187)(138,188)(139,185)(140,186)
(141,203)(142,204)(143,201)(144,202)(145,219)(146,220)(147,217)(148,218)
(149,215)(150,216)(151,213)(152,214)(153,211)(154,212)(155,209)(156,210)
(157,207)(158,208)(159,205)(160,206)(161,223)(162,224)(163,221)(164,222)
(165,239)(166,240)(167,237)(168,238)(169,235)(170,236)(171,233)(172,234)
(173,231)(174,232)(175,229)(176,230)(177,227)(178,228)(179,225)(180,226);;
s1 := (  1,125)(  2,127)(  3,126)(  4,128)(  5,121)(  6,123)(  7,122)(  8,124)
(  9,137)( 10,139)( 11,138)( 12,140)( 13,133)( 14,135)( 15,134)( 16,136)
( 17,129)( 18,131)( 19,130)( 20,132)( 21,165)( 22,167)( 23,166)( 24,168)
( 25,161)( 26,163)( 27,162)( 28,164)( 29,177)( 30,179)( 31,178)( 32,180)
( 33,173)( 34,175)( 35,174)( 36,176)( 37,169)( 38,171)( 39,170)( 40,172)
( 41,145)( 42,147)( 43,146)( 44,148)( 45,141)( 46,143)( 47,142)( 48,144)
( 49,157)( 50,159)( 51,158)( 52,160)( 53,153)( 54,155)( 55,154)( 56,156)
( 57,149)( 58,151)( 59,150)( 60,152)( 61,185)( 62,187)( 63,186)( 64,188)
( 65,181)( 66,183)( 67,182)( 68,184)( 69,197)( 70,199)( 71,198)( 72,200)
( 73,193)( 74,195)( 75,194)( 76,196)( 77,189)( 78,191)( 79,190)( 80,192)
( 81,225)( 82,227)( 83,226)( 84,228)( 85,221)( 86,223)( 87,222)( 88,224)
( 89,237)( 90,239)( 91,238)( 92,240)( 93,233)( 94,235)( 95,234)( 96,236)
( 97,229)( 98,231)( 99,230)(100,232)(101,205)(102,207)(103,206)(104,208)
(105,201)(106,203)(107,202)(108,204)(109,217)(110,219)(111,218)(112,220)
(113,213)(114,215)(115,214)(116,216)(117,209)(118,211)(119,210)(120,212);;
s2 := (  1, 41)(  2, 44)(  3, 43)(  4, 42)(  5, 45)(  6, 48)(  7, 47)(  8, 46)
(  9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 53)( 14, 56)( 15, 55)( 16, 54)
( 17, 57)( 18, 60)( 19, 59)( 20, 58)( 22, 24)( 26, 28)( 30, 32)( 34, 36)
( 38, 40)( 61,101)( 62,104)( 63,103)( 64,102)( 65,105)( 66,108)( 67,107)
( 68,106)( 69,109)( 70,112)( 71,111)( 72,110)( 73,113)( 74,116)( 75,115)
( 76,114)( 77,117)( 78,120)( 79,119)( 80,118)( 82, 84)( 86, 88)( 90, 92)
( 94, 96)( 98,100)(121,221)(122,224)(123,223)(124,222)(125,225)(126,228)
(127,227)(128,226)(129,229)(130,232)(131,231)(132,230)(133,233)(134,236)
(135,235)(136,234)(137,237)(138,240)(139,239)(140,238)(141,201)(142,204)
(143,203)(144,202)(145,205)(146,208)(147,207)(148,206)(149,209)(150,212)
(151,211)(152,210)(153,213)(154,216)(155,215)(156,214)(157,217)(158,220)
(159,219)(160,218)(161,181)(162,184)(163,183)(164,182)(165,185)(166,188)
(167,187)(168,186)(169,189)(170,192)(171,191)(172,190)(173,193)(174,196)
(175,195)(176,194)(177,197)(178,200)(179,199)(180,198);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope