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