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Polytope of Type {12,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,20}*960c
if this polytope has a name.
Group : SmallGroup(960,10994)
Rank : 3
Schlafli Type : {12,20}
Number of vertices, edges, etc : 24, 240, 40
Order of s0s1s2 : 30
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,20,2} of size 1920
Vertex Figure Of :
   {2,12,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,20}*480c
   4-fold quotients : {6,20}*240b
   5-fold quotients : {12,4}*192c
   8-fold quotients : {6,10}*120
   10-fold quotients : {6,4}*96
   20-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   24-fold quotients : {2,10}*40
   40-fold quotients : {3,4}*24, {6,2}*24
   48-fold quotients : {2,5}*20
   80-fold quotients : {3,2}*12
   120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,20}*1920c, {12,40}*1920g, {12,40}*1920h, {24,20}*1920e, {24,20}*1920f
Permutation Representation (GAP) :
s0 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120)(121,181)(122,183)(123,182)(124,184)(125,185)(126,187)
(127,186)(128,188)(129,189)(130,191)(131,190)(132,192)(133,193)(134,195)
(135,194)(136,196)(137,197)(138,199)(139,198)(140,200)(141,221)(142,223)
(143,222)(144,224)(145,225)(146,227)(147,226)(148,228)(149,229)(150,231)
(151,230)(152,232)(153,233)(154,235)(155,234)(156,236)(157,237)(158,239)
(159,238)(160,240)(161,201)(162,203)(163,202)(164,204)(165,205)(166,207)
(167,206)(168,208)(169,209)(170,211)(171,210)(172,212)(173,213)(174,215)
(175,214)(176,216)(177,217)(178,219)(179,218)(180,220);;
s1 := (  1,141)(  2,142)(  3,144)(  4,143)(  5,157)(  6,158)(  7,160)(  8,159)
(  9,153)( 10,154)( 11,156)( 12,155)( 13,149)( 14,150)( 15,152)( 16,151)
( 17,145)( 18,146)( 19,148)( 20,147)( 21,121)( 22,122)( 23,124)( 24,123)
( 25,137)( 26,138)( 27,140)( 28,139)( 29,133)( 30,134)( 31,136)( 32,135)
( 33,129)( 34,130)( 35,132)( 36,131)( 37,125)( 38,126)( 39,128)( 40,127)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,177)( 46,178)( 47,180)( 48,179)
( 49,173)( 50,174)( 51,176)( 52,175)( 53,169)( 54,170)( 55,172)( 56,171)
( 57,165)( 58,166)( 59,168)( 60,167)( 61,201)( 62,202)( 63,204)( 64,203)
( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)( 72,215)
( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)( 80,207)
( 81,181)( 82,182)( 83,184)( 84,183)( 85,197)( 86,198)( 87,200)( 88,199)
( 89,193)( 90,194)( 91,196)( 92,195)( 93,189)( 94,190)( 95,192)( 96,191)
( 97,185)( 98,186)( 99,188)(100,187)(101,221)(102,222)(103,224)(104,223)
(105,237)(106,238)(107,240)(108,239)(109,233)(110,234)(111,236)(112,235)
(113,229)(114,230)(115,232)(116,231)(117,225)(118,226)(119,228)(120,227);;
s2 := (  1,  8)(  2,  7)(  3,  6)(  4,  5)(  9, 20)( 10, 19)( 11, 18)( 12, 17)
( 13, 16)( 14, 15)( 21, 28)( 22, 27)( 23, 26)( 24, 25)( 29, 40)( 30, 39)
( 31, 38)( 32, 37)( 33, 36)( 34, 35)( 41, 48)( 42, 47)( 43, 46)( 44, 45)
( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)( 61, 68)( 62, 67)
( 63, 66)( 64, 65)( 69, 80)( 70, 79)( 71, 78)( 72, 77)( 73, 76)( 74, 75)
( 81, 88)( 82, 87)( 83, 86)( 84, 85)( 89,100)( 90, 99)( 91, 98)( 92, 97)
( 93, 96)( 94, 95)(101,108)(102,107)(103,106)(104,105)(109,120)(110,119)
(111,118)(112,117)(113,116)(114,115)(121,188)(122,187)(123,186)(124,185)
(125,184)(126,183)(127,182)(128,181)(129,200)(130,199)(131,198)(132,197)
(133,196)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)
(141,208)(142,207)(143,206)(144,205)(145,204)(146,203)(147,202)(148,201)
(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)
(157,212)(158,211)(159,210)(160,209)(161,228)(162,227)(163,226)(164,225)
(165,224)(166,223)(167,222)(168,221)(169,240)(170,239)(171,238)(172,237)
(173,236)(174,235)(175,234)(176,233)(177,232)(178,231)(179,230)(180,229);;
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, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(240)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)
( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)
( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)
( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)
( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)
( 98,119)( 99,118)(100,120)(121,181)(122,183)(123,182)(124,184)(125,185)
(126,187)(127,186)(128,188)(129,189)(130,191)(131,190)(132,192)(133,193)
(134,195)(135,194)(136,196)(137,197)(138,199)(139,198)(140,200)(141,221)
(142,223)(143,222)(144,224)(145,225)(146,227)(147,226)(148,228)(149,229)
(150,231)(151,230)(152,232)(153,233)(154,235)(155,234)(156,236)(157,237)
(158,239)(159,238)(160,240)(161,201)(162,203)(163,202)(164,204)(165,205)
(166,207)(167,206)(168,208)(169,209)(170,211)(171,210)(172,212)(173,213)
(174,215)(175,214)(176,216)(177,217)(178,219)(179,218)(180,220);
s1 := Sym(240)!(  1,141)(  2,142)(  3,144)(  4,143)(  5,157)(  6,158)(  7,160)
(  8,159)(  9,153)( 10,154)( 11,156)( 12,155)( 13,149)( 14,150)( 15,152)
( 16,151)( 17,145)( 18,146)( 19,148)( 20,147)( 21,121)( 22,122)( 23,124)
( 24,123)( 25,137)( 26,138)( 27,140)( 28,139)( 29,133)( 30,134)( 31,136)
( 32,135)( 33,129)( 34,130)( 35,132)( 36,131)( 37,125)( 38,126)( 39,128)
( 40,127)( 41,161)( 42,162)( 43,164)( 44,163)( 45,177)( 46,178)( 47,180)
( 48,179)( 49,173)( 50,174)( 51,176)( 52,175)( 53,169)( 54,170)( 55,172)
( 56,171)( 57,165)( 58,166)( 59,168)( 60,167)( 61,201)( 62,202)( 63,204)
( 64,203)( 65,217)( 66,218)( 67,220)( 68,219)( 69,213)( 70,214)( 71,216)
( 72,215)( 73,209)( 74,210)( 75,212)( 76,211)( 77,205)( 78,206)( 79,208)
( 80,207)( 81,181)( 82,182)( 83,184)( 84,183)( 85,197)( 86,198)( 87,200)
( 88,199)( 89,193)( 90,194)( 91,196)( 92,195)( 93,189)( 94,190)( 95,192)
( 96,191)( 97,185)( 98,186)( 99,188)(100,187)(101,221)(102,222)(103,224)
(104,223)(105,237)(106,238)(107,240)(108,239)(109,233)(110,234)(111,236)
(112,235)(113,229)(114,230)(115,232)(116,231)(117,225)(118,226)(119,228)
(120,227);
s2 := Sym(240)!(  1,  8)(  2,  7)(  3,  6)(  4,  5)(  9, 20)( 10, 19)( 11, 18)
( 12, 17)( 13, 16)( 14, 15)( 21, 28)( 22, 27)( 23, 26)( 24, 25)( 29, 40)
( 30, 39)( 31, 38)( 32, 37)( 33, 36)( 34, 35)( 41, 48)( 42, 47)( 43, 46)
( 44, 45)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)( 61, 68)
( 62, 67)( 63, 66)( 64, 65)( 69, 80)( 70, 79)( 71, 78)( 72, 77)( 73, 76)
( 74, 75)( 81, 88)( 82, 87)( 83, 86)( 84, 85)( 89,100)( 90, 99)( 91, 98)
( 92, 97)( 93, 96)( 94, 95)(101,108)(102,107)(103,106)(104,105)(109,120)
(110,119)(111,118)(112,117)(113,116)(114,115)(121,188)(122,187)(123,186)
(124,185)(125,184)(126,183)(127,182)(128,181)(129,200)(130,199)(131,198)
(132,197)(133,196)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)
(140,189)(141,208)(142,207)(143,206)(144,205)(145,204)(146,203)(147,202)
(148,201)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)
(156,213)(157,212)(158,211)(159,210)(160,209)(161,228)(162,227)(163,226)
(164,225)(165,224)(166,223)(167,222)(168,221)(169,240)(170,239)(171,238)
(172,237)(173,236)(174,235)(175,234)(176,233)(177,232)(178,231)(179,230)
(180,229);
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, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0 >; 
 
References : None.
to this polytope