Polytope of Type {20,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,12}*960b
if this polytope has a name.
Group : SmallGroup(960,10951)
Rank : 3
Schlafli Type : {20,12}
Number of vertices, edges, etc : 40, 240, 24
Order of s₀s₁s₂ : 60
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
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 : {10,12}*240, {20,6}*240b
   5-fold quotients : {4,12}*192b
   8-fold quotients : {10,6}*120
   10-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
   12-fold quotients : {10,4}*80
   20-fold quotients : {2,12}*48, {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
   60-fold quotients : {2,4}*16
   80-fold quotients : {2,3}*12
   120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {40,12}*1920e, {40,12}*1920f, {20,24}*1920c, {20,24}*1920d, {20,12}*1920c
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)(121,123)(122,124)(125,139)(126,140)
(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(141,143)(142,144)
(145,159)(146,160)(147,157)(148,158)(149,155)(150,156)(151,153)(152,154)
(161,163)(162,164)(165,179)(166,180)(167,177)(168,178)(169,175)(170,176)
(171,173)(172,174)(181,183)(182,184)(185,199)(186,200)(187,197)(188,198)
(189,195)(190,196)(191,193)(192,194)(201,203)(202,204)(205,219)(206,220)
(207,217)(208,218)(209,215)(210,216)(211,213)(212,214)(221,223)(222,224)
(225,239)(226,240)(227,237)(228,238)(229,235)(230,236)(231,233)(232,234);;
s1 := (  1,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 17)( 10, 19)( 11, 18)( 12, 20)
( 14, 15)( 21, 45)( 22, 47)( 23, 46)( 24, 48)( 25, 41)( 26, 43)( 27, 42)
( 28, 44)( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 53)( 34, 55)( 35, 54)
( 36, 56)( 37, 49)( 38, 51)( 39, 50)( 40, 52)( 61, 65)( 62, 67)( 63, 66)
( 64, 68)( 69, 77)( 70, 79)( 71, 78)( 72, 80)( 74, 75)( 81,105)( 82,107)
( 83,106)( 84,108)( 85,101)( 86,103)( 87,102)( 88,104)( 89,117)( 90,119)
( 91,118)( 92,120)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)( 98,111)
( 99,110)(100,112)(121,185)(122,187)(123,186)(124,188)(125,181)(126,183)
(127,182)(128,184)(129,197)(130,199)(131,198)(132,200)(133,193)(134,195)
(135,194)(136,196)(137,189)(138,191)(139,190)(140,192)(141,225)(142,227)
(143,226)(144,228)(145,221)(146,223)(147,222)(148,224)(149,237)(150,239)
(151,238)(152,240)(153,233)(154,235)(155,234)(156,236)(157,229)(158,231)
(159,230)(160,232)(161,205)(162,207)(163,206)(164,208)(165,201)(166,203)
(167,202)(168,204)(169,217)(170,219)(171,218)(172,220)(173,213)(174,215)
(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);;
s2 := (  1,161)(  2,164)(  3,163)(  4,162)(  5,165)(  6,168)(  7,167)(  8,166)
(  9,169)( 10,172)( 11,171)( 12,170)( 13,173)( 14,176)( 15,175)( 16,174)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,141)( 22,144)( 23,143)( 24,142)
( 25,145)( 26,148)( 27,147)( 28,146)( 29,149)( 30,152)( 31,151)( 32,150)
( 33,153)( 34,156)( 35,155)( 36,154)( 37,157)( 38,160)( 39,159)( 40,158)
( 41,121)( 42,124)( 43,123)( 44,122)( 45,125)( 46,128)( 47,127)( 48,126)
( 49,129)( 50,132)( 51,131)( 52,130)( 53,133)( 54,136)( 55,135)( 56,134)
( 57,137)( 58,140)( 59,139)( 60,138)( 61,221)( 62,224)( 63,223)( 64,222)
( 65,225)( 66,228)( 67,227)( 68,226)( 69,229)( 70,232)( 71,231)( 72,230)
( 73,233)( 74,236)( 75,235)( 76,234)( 77,237)( 78,240)( 79,239)( 80,238)
( 81,201)( 82,204)( 83,203)( 84,202)( 85,205)( 86,208)( 87,207)( 88,206)
( 89,209)( 90,212)( 91,211)( 92,210)( 93,213)( 94,216)( 95,215)( 96,214)
( 97,217)( 98,220)( 99,219)(100,218)(101,181)(102,184)(103,183)(104,182)
(105,185)(106,188)(107,187)(108,186)(109,189)(110,192)(111,191)(112,190)
(113,193)(114,196)(115,195)(116,194)(117,197)(118,200)(119,199)(120,198);;
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, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*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 ];;
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)(121,123)(122,124)(125,139)
(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(141,143)
(142,144)(145,159)(146,160)(147,157)(148,158)(149,155)(150,156)(151,153)
(152,154)(161,163)(162,164)(165,179)(166,180)(167,177)(168,178)(169,175)
(170,176)(171,173)(172,174)(181,183)(182,184)(185,199)(186,200)(187,197)
(188,198)(189,195)(190,196)(191,193)(192,194)(201,203)(202,204)(205,219)
(206,220)(207,217)(208,218)(209,215)(210,216)(211,213)(212,214)(221,223)
(222,224)(225,239)(226,240)(227,237)(228,238)(229,235)(230,236)(231,233)
(232,234);
s1 := Sym(240)!(  1,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 17)( 10, 19)( 11, 18)
( 12, 20)( 14, 15)( 21, 45)( 22, 47)( 23, 46)( 24, 48)( 25, 41)( 26, 43)
( 27, 42)( 28, 44)( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 53)( 34, 55)
( 35, 54)( 36, 56)( 37, 49)( 38, 51)( 39, 50)( 40, 52)( 61, 65)( 62, 67)
( 63, 66)( 64, 68)( 69, 77)( 70, 79)( 71, 78)( 72, 80)( 74, 75)( 81,105)
( 82,107)( 83,106)( 84,108)( 85,101)( 86,103)( 87,102)( 88,104)( 89,117)
( 90,119)( 91,118)( 92,120)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)
( 98,111)( 99,110)(100,112)(121,185)(122,187)(123,186)(124,188)(125,181)
(126,183)(127,182)(128,184)(129,197)(130,199)(131,198)(132,200)(133,193)
(134,195)(135,194)(136,196)(137,189)(138,191)(139,190)(140,192)(141,225)
(142,227)(143,226)(144,228)(145,221)(146,223)(147,222)(148,224)(149,237)
(150,239)(151,238)(152,240)(153,233)(154,235)(155,234)(156,236)(157,229)
(158,231)(159,230)(160,232)(161,205)(162,207)(163,206)(164,208)(165,201)
(166,203)(167,202)(168,204)(169,217)(170,219)(171,218)(172,220)(173,213)
(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);
s2 := Sym(240)!(  1,161)(  2,164)(  3,163)(  4,162)(  5,165)(  6,168)(  7,167)
(  8,166)(  9,169)( 10,172)( 11,171)( 12,170)( 13,173)( 14,176)( 15,175)
( 16,174)( 17,177)( 18,180)( 19,179)( 20,178)( 21,141)( 22,144)( 23,143)
( 24,142)( 25,145)( 26,148)( 27,147)( 28,146)( 29,149)( 30,152)( 31,151)
( 32,150)( 33,153)( 34,156)( 35,155)( 36,154)( 37,157)( 38,160)( 39,159)
( 40,158)( 41,121)( 42,124)( 43,123)( 44,122)( 45,125)( 46,128)( 47,127)
( 48,126)( 49,129)( 50,132)( 51,131)( 52,130)( 53,133)( 54,136)( 55,135)
( 56,134)( 57,137)( 58,140)( 59,139)( 60,138)( 61,221)( 62,224)( 63,223)
( 64,222)( 65,225)( 66,228)( 67,227)( 68,226)( 69,229)( 70,232)( 71,231)
( 72,230)( 73,233)( 74,236)( 75,235)( 76,234)( 77,237)( 78,240)( 79,239)
( 80,238)( 81,201)( 82,204)( 83,203)( 84,202)( 85,205)( 86,208)( 87,207)
( 88,206)( 89,209)( 90,212)( 91,211)( 92,210)( 93,213)( 94,216)( 95,215)
( 96,214)( 97,217)( 98,220)( 99,219)(100,218)(101,181)(102,184)(103,183)
(104,182)(105,185)(106,188)(107,187)(108,186)(109,189)(110,192)(111,191)
(112,190)(113,193)(114,196)(115,195)(116,194)(117,197)(118,200)(119,199)
(120,198);
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, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*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 >;

References : None.
to this polytope