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Polytope of Type {21,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6,4}*1008
if this polytope has a name.
Group : SmallGroup(1008,797)
Rank : 4
Schlafli Type : {21,6,4}
Number of vertices, edges, etc : 21, 63, 12, 4
Order of s0s1s2s3 : 84
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 : {21,6,2}*504
   3-fold quotients : {21,2,4}*336
   6-fold quotients : {21,2,2}*168
   7-fold quotients : {3,6,4}*144
   9-fold quotients : {7,2,4}*112
   14-fold quotients : {3,6,2}*72
   18-fold quotients : {7,2,2}*56
   21-fold quotients : {3,2,4}*48
   42-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)
( 13, 17)( 14, 16)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)( 27, 45)
( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)
( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 65, 70)
( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)
( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)
(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(128,133)(129,132)
(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)
(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,183)
(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,176)(163,182)
(164,181)(165,180)(166,179)(167,178)(168,177)(191,196)(192,195)(193,194)
(197,204)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(211,232)
(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,246)(219,252)
(220,251)(221,250)(222,249)(223,248)(224,247)(225,239)(226,245)(227,244)
(228,243)(229,242)(230,241)(231,240);;
s1 := (  1, 30)(  2, 29)(  3, 35)(  4, 34)(  5, 33)(  6, 32)(  7, 31)(  8, 23)
(  9, 22)( 10, 28)( 11, 27)( 12, 26)( 13, 25)( 14, 24)( 15, 37)( 16, 36)
( 17, 42)( 18, 41)( 19, 40)( 20, 39)( 21, 38)( 43, 51)( 44, 50)( 45, 56)
( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 93)
( 65, 92)( 66, 98)( 67, 97)( 68, 96)( 69, 95)( 70, 94)( 71, 86)( 72, 85)
( 73, 91)( 74, 90)( 75, 89)( 76, 88)( 77, 87)( 78,100)( 79, 99)( 80,105)
( 81,104)( 82,103)( 83,102)( 84,101)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,121)(122,126)(123,125)(127,156)(128,155)
(129,161)(130,160)(131,159)(132,158)(133,157)(134,149)(135,148)(136,154)
(137,153)(138,152)(139,151)(140,150)(141,163)(142,162)(143,168)(144,167)
(145,166)(146,165)(147,164)(169,177)(170,176)(171,182)(172,181)(173,180)
(174,179)(175,178)(183,184)(185,189)(186,188)(190,219)(191,218)(192,224)
(193,223)(194,222)(195,221)(196,220)(197,212)(198,211)(199,217)(200,216)
(201,215)(202,214)(203,213)(204,226)(205,225)(206,231)(207,230)(208,229)
(209,228)(210,227)(232,240)(233,239)(234,245)(235,244)(236,243)(237,242)
(238,241)(246,247)(248,252)(249,251);;
s2 := ( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)
( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)
( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 85,106)( 86,107)( 87,108)
( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)( 95,116)
( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)(103,124)
(104,125)(105,126)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)
(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)
(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,232)
(149,233)(150,234)(151,235)(152,236)(153,237)(154,238)(155,239)(156,240)
(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)(164,248)
(165,249)(166,250)(167,251)(168,252)(169,211)(170,212)(171,213)(172,214)
(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)
(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)
(189,231);;
s3 := (  1,127)(  2,128)(  3,129)(  4,130)(  5,131)(  6,132)(  7,133)(  8,134)
(  9,135)( 10,136)( 11,137)( 12,138)( 13,139)( 14,140)( 15,141)( 16,142)
( 17,143)( 18,144)( 19,145)( 20,146)( 21,147)( 22,148)( 23,149)( 24,150)
( 25,151)( 26,152)( 27,153)( 28,154)( 29,155)( 30,156)( 31,157)( 32,158)
( 33,159)( 34,160)( 35,161)( 36,162)( 37,163)( 38,164)( 39,165)( 40,166)
( 41,167)( 42,168)( 43,169)( 44,170)( 45,171)( 46,172)( 47,173)( 48,174)
( 49,175)( 50,176)( 51,177)( 52,178)( 53,179)( 54,180)( 55,181)( 56,182)
( 57,183)( 58,184)( 59,185)( 60,186)( 61,187)( 62,188)( 63,189)( 64,190)
( 65,191)( 66,192)( 67,193)( 68,194)( 69,195)( 70,196)( 71,197)( 72,198)
( 73,199)( 74,200)( 75,201)( 76,202)( 77,203)( 78,204)( 79,205)( 80,206)
( 81,207)( 82,208)( 83,209)( 84,210)( 85,211)( 86,212)( 87,213)( 88,214)
( 89,215)( 90,216)( 91,217)( 92,218)( 93,219)( 94,220)( 95,221)( 96,222)
( 97,223)( 98,224)( 99,225)(100,226)(101,227)(102,228)(103,229)(104,230)
(105,231)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)
(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)(120,246)
(121,247)(122,248)(123,249)(124,250)(125,251)(126,252);;
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(252)!(  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)
( 12, 18)( 13, 17)( 14, 16)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)
( 27, 45)( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)
( 35, 58)( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)
( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)
( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)
( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)
( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(128,133)
(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)
(140,142)(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)
(155,183)(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,176)
(163,182)(164,181)(165,180)(166,179)(167,178)(168,177)(191,196)(192,195)
(193,194)(197,204)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)
(211,232)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,246)
(219,252)(220,251)(221,250)(222,249)(223,248)(224,247)(225,239)(226,245)
(227,244)(228,243)(229,242)(230,241)(231,240);
s1 := Sym(252)!(  1, 30)(  2, 29)(  3, 35)(  4, 34)(  5, 33)(  6, 32)(  7, 31)
(  8, 23)(  9, 22)( 10, 28)( 11, 27)( 12, 26)( 13, 25)( 14, 24)( 15, 37)
( 16, 36)( 17, 42)( 18, 41)( 19, 40)( 20, 39)( 21, 38)( 43, 51)( 44, 50)
( 45, 56)( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)
( 64, 93)( 65, 92)( 66, 98)( 67, 97)( 68, 96)( 69, 95)( 70, 94)( 71, 86)
( 72, 85)( 73, 91)( 74, 90)( 75, 89)( 76, 88)( 77, 87)( 78,100)( 79, 99)
( 80,105)( 81,104)( 82,103)( 83,102)( 84,101)(106,114)(107,113)(108,119)
(109,118)(110,117)(111,116)(112,115)(120,121)(122,126)(123,125)(127,156)
(128,155)(129,161)(130,160)(131,159)(132,158)(133,157)(134,149)(135,148)
(136,154)(137,153)(138,152)(139,151)(140,150)(141,163)(142,162)(143,168)
(144,167)(145,166)(146,165)(147,164)(169,177)(170,176)(171,182)(172,181)
(173,180)(174,179)(175,178)(183,184)(185,189)(186,188)(190,219)(191,218)
(192,224)(193,223)(194,222)(195,221)(196,220)(197,212)(198,211)(199,217)
(200,216)(201,215)(202,214)(203,213)(204,226)(205,225)(206,231)(207,230)
(208,229)(209,228)(210,227)(232,240)(233,239)(234,245)(235,244)(236,243)
(237,242)(238,241)(246,247)(248,252)(249,251);
s2 := Sym(252)!( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)
( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)
( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 85,106)( 86,107)
( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)
( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)
(103,124)(104,125)(105,126)(127,190)(128,191)(129,192)(130,193)(131,194)
(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)
(140,203)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)
(148,232)(149,233)(150,234)(151,235)(152,236)(153,237)(154,238)(155,239)
(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)
(164,248)(165,249)(166,250)(167,251)(168,252)(169,211)(170,212)(171,213)
(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)
(180,222)(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)
(188,230)(189,231);
s3 := Sym(252)!(  1,127)(  2,128)(  3,129)(  4,130)(  5,131)(  6,132)(  7,133)
(  8,134)(  9,135)( 10,136)( 11,137)( 12,138)( 13,139)( 14,140)( 15,141)
( 16,142)( 17,143)( 18,144)( 19,145)( 20,146)( 21,147)( 22,148)( 23,149)
( 24,150)( 25,151)( 26,152)( 27,153)( 28,154)( 29,155)( 30,156)( 31,157)
( 32,158)( 33,159)( 34,160)( 35,161)( 36,162)( 37,163)( 38,164)( 39,165)
( 40,166)( 41,167)( 42,168)( 43,169)( 44,170)( 45,171)( 46,172)( 47,173)
( 48,174)( 49,175)( 50,176)( 51,177)( 52,178)( 53,179)( 54,180)( 55,181)
( 56,182)( 57,183)( 58,184)( 59,185)( 60,186)( 61,187)( 62,188)( 63,189)
( 64,190)( 65,191)( 66,192)( 67,193)( 68,194)( 69,195)( 70,196)( 71,197)
( 72,198)( 73,199)( 74,200)( 75,201)( 76,202)( 77,203)( 78,204)( 79,205)
( 80,206)( 81,207)( 82,208)( 83,209)( 84,210)( 85,211)( 86,212)( 87,213)
( 88,214)( 89,215)( 90,216)( 91,217)( 92,218)( 93,219)( 94,220)( 95,221)
( 96,222)( 97,223)( 98,224)( 99,225)(100,226)(101,227)(102,228)(103,229)
(104,230)(105,231)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)
(112,238)(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)
(120,246)(121,247)(122,248)(123,249)(124,250)(125,251)(126,252);
poly := sub<Sym(252)|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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope