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Polytope of Type {2,20,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,20,24}*1920a
if this polytope has a name.
Group : SmallGroup(1920,148923)
Rank : 4
Schlafli Type : {2,20,24}
Number of vertices, edges, etc : 2, 20, 240, 24
Order of s0s1s2s3 : 120
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,10,24}*960, {2,20,12}*960
   3-fold quotients : {2,20,8}*640a
   4-fold quotients : {2,10,12}*480, {2,20,6}*480a
   5-fold quotients : {2,4,24}*384a
   6-fold quotients : {2,20,4}*320, {2,10,8}*320
   8-fold quotients : {2,10,6}*240
   10-fold quotients : {2,4,12}*192a, {2,2,24}*192
   12-fold quotients : {2,20,2}*160, {2,10,4}*160
   15-fold quotients : {2,4,8}*128a
   20-fold quotients : {2,2,12}*96, {2,4,6}*96a
   24-fold quotients : {2,10,2}*80
   30-fold quotients : {2,4,4}*64, {2,2,8}*64
   40-fold quotients : {2,2,6}*48
   48-fold quotients : {2,5,2}*40
   60-fold quotients : {2,2,4}*32, {2,4,2}*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 := (1,2);;
s1 := (  4,  7)(  5,  6)(  9, 12)( 10, 11)( 14, 17)( 15, 16)( 19, 22)( 20, 21)
( 24, 27)( 25, 26)( 29, 32)( 30, 31)( 34, 37)( 35, 36)( 39, 42)( 40, 41)
( 44, 47)( 45, 46)( 49, 52)( 50, 51)( 54, 57)( 55, 56)( 59, 62)( 60, 61)
( 64, 67)( 65, 66)( 69, 72)( 70, 71)( 74, 77)( 75, 76)( 79, 82)( 80, 81)
( 84, 87)( 85, 86)( 89, 92)( 90, 91)( 94, 97)( 95, 96)( 99,102)(100,101)
(104,107)(105,106)(109,112)(110,111)(114,117)(115,116)(119,122)(120,121)
(123,183)(124,187)(125,186)(126,185)(127,184)(128,188)(129,192)(130,191)
(131,190)(132,189)(133,193)(134,197)(135,196)(136,195)(137,194)(138,198)
(139,202)(140,201)(141,200)(142,199)(143,203)(144,207)(145,206)(146,205)
(147,204)(148,208)(149,212)(150,211)(151,210)(152,209)(153,213)(154,217)
(155,216)(156,215)(157,214)(158,218)(159,222)(160,221)(161,220)(162,219)
(163,223)(164,227)(165,226)(166,225)(167,224)(168,228)(169,232)(170,231)
(171,230)(172,229)(173,233)(174,237)(175,236)(176,235)(177,234)(178,238)
(179,242)(180,241)(181,240)(182,239);;
s2 := (  3,124)(  4,123)(  5,127)(  6,126)(  7,125)(  8,134)(  9,133)( 10,137)
( 11,136)( 12,135)( 13,129)( 14,128)( 15,132)( 16,131)( 17,130)( 18,139)
( 19,138)( 20,142)( 21,141)( 22,140)( 23,149)( 24,148)( 25,152)( 26,151)
( 27,150)( 28,144)( 29,143)( 30,147)( 31,146)( 32,145)( 33,169)( 34,168)
( 35,172)( 36,171)( 37,170)( 38,179)( 39,178)( 40,182)( 41,181)( 42,180)
( 43,174)( 44,173)( 45,177)( 46,176)( 47,175)( 48,154)( 49,153)( 50,157)
( 51,156)( 52,155)( 53,164)( 54,163)( 55,167)( 56,166)( 57,165)( 58,159)
( 59,158)( 60,162)( 61,161)( 62,160)( 63,184)( 64,183)( 65,187)( 66,186)
( 67,185)( 68,194)( 69,193)( 70,197)( 71,196)( 72,195)( 73,189)( 74,188)
( 75,192)( 76,191)( 77,190)( 78,199)( 79,198)( 80,202)( 81,201)( 82,200)
( 83,209)( 84,208)( 85,212)( 86,211)( 87,210)( 88,204)( 89,203)( 90,207)
( 91,206)( 92,205)( 93,229)( 94,228)( 95,232)( 96,231)( 97,230)( 98,239)
( 99,238)(100,242)(101,241)(102,240)(103,234)(104,233)(105,237)(106,236)
(107,235)(108,214)(109,213)(110,217)(111,216)(112,215)(113,224)(114,223)
(115,227)(116,226)(117,225)(118,219)(119,218)(120,222)(121,221)(122,220);;
s3 := (  3,  8)(  4,  9)(  5, 10)(  6, 11)(  7, 12)( 18, 23)( 19, 24)( 20, 25)
( 21, 26)( 22, 27)( 33, 53)( 34, 54)( 35, 55)( 36, 56)( 37, 57)( 38, 48)
( 39, 49)( 40, 50)( 41, 51)( 42, 52)( 43, 58)( 44, 59)( 45, 60)( 46, 61)
( 47, 62)( 63, 68)( 64, 69)( 65, 70)( 66, 71)( 67, 72)( 78, 83)( 79, 84)
( 80, 85)( 81, 86)( 82, 87)( 93,113)( 94,114)( 95,115)( 96,116)( 97,117)
( 98,108)( 99,109)(100,110)(101,111)(102,112)(103,118)(104,119)(105,120)
(106,121)(107,122)(123,158)(124,159)(125,160)(126,161)(127,162)(128,153)
(129,154)(130,155)(131,156)(132,157)(133,163)(134,164)(135,165)(136,166)
(137,167)(138,173)(139,174)(140,175)(141,176)(142,177)(143,168)(144,169)
(145,170)(146,171)(147,172)(148,178)(149,179)(150,180)(151,181)(152,182)
(183,218)(184,219)(185,220)(186,221)(187,222)(188,213)(189,214)(190,215)
(191,216)(192,217)(193,223)(194,224)(195,225)(196,226)(197,227)(198,233)
(199,234)(200,235)(201,236)(202,237)(203,228)(204,229)(205,230)(206,231)
(207,232)(208,238)(209,239)(210,240)(211,241)(212,242);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(242)!(1,2);
s1 := Sym(242)!(  4,  7)(  5,  6)(  9, 12)( 10, 11)( 14, 17)( 15, 16)( 19, 22)
( 20, 21)( 24, 27)( 25, 26)( 29, 32)( 30, 31)( 34, 37)( 35, 36)( 39, 42)
( 40, 41)( 44, 47)( 45, 46)( 49, 52)( 50, 51)( 54, 57)( 55, 56)( 59, 62)
( 60, 61)( 64, 67)( 65, 66)( 69, 72)( 70, 71)( 74, 77)( 75, 76)( 79, 82)
( 80, 81)( 84, 87)( 85, 86)( 89, 92)( 90, 91)( 94, 97)( 95, 96)( 99,102)
(100,101)(104,107)(105,106)(109,112)(110,111)(114,117)(115,116)(119,122)
(120,121)(123,183)(124,187)(125,186)(126,185)(127,184)(128,188)(129,192)
(130,191)(131,190)(132,189)(133,193)(134,197)(135,196)(136,195)(137,194)
(138,198)(139,202)(140,201)(141,200)(142,199)(143,203)(144,207)(145,206)
(146,205)(147,204)(148,208)(149,212)(150,211)(151,210)(152,209)(153,213)
(154,217)(155,216)(156,215)(157,214)(158,218)(159,222)(160,221)(161,220)
(162,219)(163,223)(164,227)(165,226)(166,225)(167,224)(168,228)(169,232)
(170,231)(171,230)(172,229)(173,233)(174,237)(175,236)(176,235)(177,234)
(178,238)(179,242)(180,241)(181,240)(182,239);
s2 := Sym(242)!(  3,124)(  4,123)(  5,127)(  6,126)(  7,125)(  8,134)(  9,133)
( 10,137)( 11,136)( 12,135)( 13,129)( 14,128)( 15,132)( 16,131)( 17,130)
( 18,139)( 19,138)( 20,142)( 21,141)( 22,140)( 23,149)( 24,148)( 25,152)
( 26,151)( 27,150)( 28,144)( 29,143)( 30,147)( 31,146)( 32,145)( 33,169)
( 34,168)( 35,172)( 36,171)( 37,170)( 38,179)( 39,178)( 40,182)( 41,181)
( 42,180)( 43,174)( 44,173)( 45,177)( 46,176)( 47,175)( 48,154)( 49,153)
( 50,157)( 51,156)( 52,155)( 53,164)( 54,163)( 55,167)( 56,166)( 57,165)
( 58,159)( 59,158)( 60,162)( 61,161)( 62,160)( 63,184)( 64,183)( 65,187)
( 66,186)( 67,185)( 68,194)( 69,193)( 70,197)( 71,196)( 72,195)( 73,189)
( 74,188)( 75,192)( 76,191)( 77,190)( 78,199)( 79,198)( 80,202)( 81,201)
( 82,200)( 83,209)( 84,208)( 85,212)( 86,211)( 87,210)( 88,204)( 89,203)
( 90,207)( 91,206)( 92,205)( 93,229)( 94,228)( 95,232)( 96,231)( 97,230)
( 98,239)( 99,238)(100,242)(101,241)(102,240)(103,234)(104,233)(105,237)
(106,236)(107,235)(108,214)(109,213)(110,217)(111,216)(112,215)(113,224)
(114,223)(115,227)(116,226)(117,225)(118,219)(119,218)(120,222)(121,221)
(122,220);
s3 := Sym(242)!(  3,  8)(  4,  9)(  5, 10)(  6, 11)(  7, 12)( 18, 23)( 19, 24)
( 20, 25)( 21, 26)( 22, 27)( 33, 53)( 34, 54)( 35, 55)( 36, 56)( 37, 57)
( 38, 48)( 39, 49)( 40, 50)( 41, 51)( 42, 52)( 43, 58)( 44, 59)( 45, 60)
( 46, 61)( 47, 62)( 63, 68)( 64, 69)( 65, 70)( 66, 71)( 67, 72)( 78, 83)
( 79, 84)( 80, 85)( 81, 86)( 82, 87)( 93,113)( 94,114)( 95,115)( 96,116)
( 97,117)( 98,108)( 99,109)(100,110)(101,111)(102,112)(103,118)(104,119)
(105,120)(106,121)(107,122)(123,158)(124,159)(125,160)(126,161)(127,162)
(128,153)(129,154)(130,155)(131,156)(132,157)(133,163)(134,164)(135,165)
(136,166)(137,167)(138,173)(139,174)(140,175)(141,176)(142,177)(143,168)
(144,169)(145,170)(146,171)(147,172)(148,178)(149,179)(150,180)(151,181)
(152,182)(183,218)(184,219)(185,220)(186,221)(187,222)(188,213)(189,214)
(190,215)(191,216)(192,217)(193,223)(194,224)(195,225)(196,226)(197,227)
(198,233)(199,234)(200,235)(201,236)(202,237)(203,228)(204,229)(205,230)
(206,231)(207,232)(208,238)(209,239)(210,240)(211,241)(212,242);
poly := sub<Sym(242)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

to this polytope