Polytope of Type {20,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4,6}*1920a
if this polytope has a name.
Group : SmallGroup(1920,151308)
Rank : 4
Schlafli Type : {20,4,6}
Number of vertices, edges, etc : 40, 80, 24, 6
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,4,6}*960
   3-fold quotients : {20,4,2}*640
   4-fold quotients : {20,2,6}*480, {10,4,6}*480
   5-fold quotients : {4,4,6}*384a
   6-fold quotients : {20,4,2}*320
   8-fold quotients : {20,2,3}*240, {10,2,6}*240
   10-fold quotients : {4,4,6}*192
   12-fold quotients : {20,2,2}*160, {10,4,2}*160
   15-fold quotients : {4,4,2}*128
   16-fold quotients : {5,2,6}*120, {10,2,3}*120
   20-fold quotients : {2,4,6}*96a, {4,2,6}*96
   24-fold quotients : {10,2,2}*80
   30-fold quotients : {4,4,2}*64
   32-fold quotients : {5,2,3}*60
   40-fold quotients : {4,2,3}*48, {2,2,6}*48
   48-fold quotients : {5,2,2}*40
   60-fold quotients : {2,4,2}*32, {4,2,2}*32
   80-fold quotients : {2,2,3}*24
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,121)(  2,125)(  3,124)(  4,123)(  5,122)(  6,126)(  7,130)(  8,129)
(  9,128)( 10,127)( 11,131)( 12,135)( 13,134)( 14,133)( 15,132)( 16,136)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,141)( 22,145)( 23,144)( 24,143)
( 25,142)( 26,146)( 27,150)( 28,149)( 29,148)( 30,147)( 31,151)( 32,155)
( 33,154)( 34,153)( 35,152)( 36,156)( 37,160)( 38,159)( 39,158)( 40,157)
( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)( 48,169)
( 49,168)( 50,167)( 51,171)( 52,175)( 53,174)( 54,173)( 55,172)( 56,176)
( 57,180)( 58,179)( 59,178)( 60,177)( 61,181)( 62,185)( 63,184)( 64,183)
( 65,182)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,191)( 72,195)
( 73,194)( 74,193)( 75,192)( 76,196)( 77,200)( 78,199)( 79,198)( 80,197)
( 81,201)( 82,205)( 83,204)( 84,203)( 85,202)( 86,206)( 87,210)( 88,209)
( 89,208)( 90,207)( 91,211)( 92,215)( 93,214)( 94,213)( 95,212)( 96,216)
( 97,220)( 98,219)( 99,218)(100,217)(101,221)(102,225)(103,224)(104,223)
(105,222)(106,226)(107,230)(108,229)(109,228)(110,227)(111,231)(112,235)
(113,234)(114,233)(115,232)(116,236)(117,240)(118,239)(119,238)(120,237);;
s1 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)
(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)(118,120)
(121,152)(122,151)(123,155)(124,154)(125,153)(126,157)(127,156)(128,160)
(129,159)(130,158)(131,162)(132,161)(133,165)(134,164)(135,163)(136,167)
(137,166)(138,170)(139,169)(140,168)(141,172)(142,171)(143,175)(144,174)
(145,173)(146,177)(147,176)(148,180)(149,179)(150,178)(181,212)(182,211)
(183,215)(184,214)(185,213)(186,217)(187,216)(188,220)(189,219)(190,218)
(191,222)(192,221)(193,225)(194,224)(195,223)(196,227)(197,226)(198,230)
(199,229)(200,228)(201,232)(202,231)(203,235)(204,234)(205,233)(206,237)
(207,236)(208,240)(209,239)(210,238);;
s2 := (  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 71)(  7, 72)(  8, 73)
(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)( 16, 76)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 86)( 22, 87)( 23, 88)( 24, 89)
( 25, 90)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46, 91)( 47, 92)( 48, 93)
( 49, 94)( 50, 95)( 51,101)( 52,102)( 53,103)( 54,104)( 55,105)( 56, 96)
( 57, 97)( 58, 98)( 59, 99)( 60,100)(121,181)(122,182)(123,183)(124,184)
(125,185)(126,191)(127,192)(128,193)(129,194)(130,195)(131,186)(132,187)
(133,188)(134,189)(135,190)(136,196)(137,197)(138,198)(139,199)(140,200)
(141,206)(142,207)(143,208)(144,209)(145,210)(146,201)(147,202)(148,203)
(149,204)(150,205)(151,226)(152,227)(153,228)(154,229)(155,230)(156,236)
(157,237)(158,238)(159,239)(160,240)(161,231)(162,232)(163,233)(164,234)
(165,235)(166,211)(167,212)(168,213)(169,214)(170,215)(171,221)(172,222)
(173,223)(174,224)(175,225)(176,216)(177,217)(178,218)(179,219)(180,220);;
s3 := (  1, 66)(  2, 67)(  3, 68)(  4, 69)(  5, 70)(  6, 61)(  7, 62)(  8, 63)
(  9, 64)( 10, 65)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)( 16, 81)
( 17, 82)( 18, 83)( 19, 84)( 20, 85)( 21, 76)( 22, 77)( 23, 78)( 24, 79)
( 25, 80)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 96)( 32, 97)
( 33, 98)( 34, 99)( 35,100)( 36, 91)( 37, 92)( 38, 93)( 39, 94)( 40, 95)
( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,111)( 47,112)( 48,113)
( 49,114)( 50,115)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)( 56,116)
( 57,117)( 58,118)( 59,119)( 60,120)(121,186)(122,187)(123,188)(124,189)
(125,190)(126,181)(127,182)(128,183)(129,184)(130,185)(131,191)(132,192)
(133,193)(134,194)(135,195)(136,201)(137,202)(138,203)(139,204)(140,205)
(141,196)(142,197)(143,198)(144,199)(145,200)(146,206)(147,207)(148,208)
(149,209)(150,210)(151,216)(152,217)(153,218)(154,219)(155,220)(156,211)
(157,212)(158,213)(159,214)(160,215)(161,221)(162,222)(163,223)(164,224)
(165,225)(166,231)(167,232)(168,233)(169,234)(170,235)(171,226)(172,227)
(173,228)(174,229)(175,230)(176,236)(177,237)(178,238)(179,239)(180,240);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(  1,121)(  2,125)(  3,124)(  4,123)(  5,122)(  6,126)(  7,130)
(  8,129)(  9,128)( 10,127)( 11,131)( 12,135)( 13,134)( 14,133)( 15,132)
( 16,136)( 17,140)( 18,139)( 19,138)( 20,137)( 21,141)( 22,145)( 23,144)
( 24,143)( 25,142)( 26,146)( 27,150)( 28,149)( 29,148)( 30,147)( 31,151)
( 32,155)( 33,154)( 34,153)( 35,152)( 36,156)( 37,160)( 38,159)( 39,158)
( 40,157)( 41,161)( 42,165)( 43,164)( 44,163)( 45,162)( 46,166)( 47,170)
( 48,169)( 49,168)( 50,167)( 51,171)( 52,175)( 53,174)( 54,173)( 55,172)
( 56,176)( 57,180)( 58,179)( 59,178)( 60,177)( 61,181)( 62,185)( 63,184)
( 64,183)( 65,182)( 66,186)( 67,190)( 68,189)( 69,188)( 70,187)( 71,191)
( 72,195)( 73,194)( 74,193)( 75,192)( 76,196)( 77,200)( 78,199)( 79,198)
( 80,197)( 81,201)( 82,205)( 83,204)( 84,203)( 85,202)( 86,206)( 87,210)
( 88,209)( 89,208)( 90,207)( 91,211)( 92,215)( 93,214)( 94,213)( 95,212)
( 96,216)( 97,220)( 98,219)( 99,218)(100,217)(101,221)(102,225)(103,224)
(104,223)(105,222)(106,226)(107,230)(108,229)(109,228)(110,227)(111,231)
(112,235)(113,234)(114,233)(115,232)(116,236)(117,240)(118,239)(119,238)
(120,237);
s1 := Sym(240)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)
( 98,100)(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)
(118,120)(121,152)(122,151)(123,155)(124,154)(125,153)(126,157)(127,156)
(128,160)(129,159)(130,158)(131,162)(132,161)(133,165)(134,164)(135,163)
(136,167)(137,166)(138,170)(139,169)(140,168)(141,172)(142,171)(143,175)
(144,174)(145,173)(146,177)(147,176)(148,180)(149,179)(150,178)(181,212)
(182,211)(183,215)(184,214)(185,213)(186,217)(187,216)(188,220)(189,219)
(190,218)(191,222)(192,221)(193,225)(194,224)(195,223)(196,227)(197,226)
(198,230)(199,229)(200,228)(201,232)(202,231)(203,235)(204,234)(205,233)
(206,237)(207,236)(208,240)(209,239)(210,238);
s2 := Sym(240)!(  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 71)(  7, 72)
(  8, 73)(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)
( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 86)( 22, 87)( 23, 88)
( 24, 89)( 25, 90)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46, 91)( 47, 92)
( 48, 93)( 49, 94)( 50, 95)( 51,101)( 52,102)( 53,103)( 54,104)( 55,105)
( 56, 96)( 57, 97)( 58, 98)( 59, 99)( 60,100)(121,181)(122,182)(123,183)
(124,184)(125,185)(126,191)(127,192)(128,193)(129,194)(130,195)(131,186)
(132,187)(133,188)(134,189)(135,190)(136,196)(137,197)(138,198)(139,199)
(140,200)(141,206)(142,207)(143,208)(144,209)(145,210)(146,201)(147,202)
(148,203)(149,204)(150,205)(151,226)(152,227)(153,228)(154,229)(155,230)
(156,236)(157,237)(158,238)(159,239)(160,240)(161,231)(162,232)(163,233)
(164,234)(165,235)(166,211)(167,212)(168,213)(169,214)(170,215)(171,221)
(172,222)(173,223)(174,224)(175,225)(176,216)(177,217)(178,218)(179,219)
(180,220);
s3 := Sym(240)!(  1, 66)(  2, 67)(  3, 68)(  4, 69)(  5, 70)(  6, 61)(  7, 62)
(  8, 63)(  9, 64)( 10, 65)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)
( 16, 81)( 17, 82)( 18, 83)( 19, 84)( 20, 85)( 21, 76)( 22, 77)( 23, 78)
( 24, 79)( 25, 80)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 96)
( 32, 97)( 33, 98)( 34, 99)( 35,100)( 36, 91)( 37, 92)( 38, 93)( 39, 94)
( 40, 95)( 41,101)( 42,102)( 43,103)( 44,104)( 45,105)( 46,111)( 47,112)
( 48,113)( 49,114)( 50,115)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)
( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)(121,186)(122,187)(123,188)
(124,189)(125,190)(126,181)(127,182)(128,183)(129,184)(130,185)(131,191)
(132,192)(133,193)(134,194)(135,195)(136,201)(137,202)(138,203)(139,204)
(140,205)(141,196)(142,197)(143,198)(144,199)(145,200)(146,206)(147,207)
(148,208)(149,209)(150,210)(151,216)(152,217)(153,218)(154,219)(155,220)
(156,211)(157,212)(158,213)(159,214)(160,215)(161,221)(162,222)(163,223)
(164,224)(165,225)(166,231)(167,232)(168,233)(169,234)(170,235)(171,226)
(172,227)(173,228)(174,229)(175,230)(176,236)(177,237)(178,238)(179,239)
(180,240);
poly := sub<Sym(240)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope