Polytope of Type {10,4,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,6,3}*1920
Also Known As : {{10,4|2},{4,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(1920,240151)
Rank : 5
Schlafli Type : {10,4,6,3}
Number of vertices, edges, etc : 10, 20, 16, 12, 4
Order of s0s1s2s3s4 : 20
Order of s0s1s2s3s4s3s2s1 : 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 : {10,2,6,3}*960
4-fold quotients : {5,2,6,3}*480, {10,2,3,3}*480
5-fold quotients : {2,4,6,3}*384
8-fold quotients : {5,2,3,3}*240
10-fold quotients : {2,2,6,3}*192
20-fold quotients : {2,2,3,3}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 17)( 6, 18)( 7, 19)( 8, 20)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 33)( 30, 34)( 31, 35)( 32, 36)
( 45, 57)( 46, 58)( 47, 59)( 48, 60)( 49, 53)( 50, 54)( 51, 55)( 52, 56)
( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)(112,116)
(125,137)(126,138)(127,139)(128,140)(129,133)(130,134)(131,135)(132,136)
(145,157)(146,158)(147,159)(148,160)(149,153)(150,154)(151,155)(152,156)
(165,177)(166,178)(167,179)(168,180)(169,173)(170,174)(171,175)(172,176)
(185,197)(186,198)(187,199)(188,200)(189,193)(190,194)(191,195)(192,196)
(205,217)(206,218)(207,219)(208,220)(209,213)(210,214)(211,215)(212,216)
(225,237)(226,238)(227,239)(228,240)(229,233)(230,234)(231,235)(232,236);;
s1 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 17)( 10, 18)( 11, 19)( 12, 20)
( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 29, 37)( 30, 38)( 31, 39)( 32, 40)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)
( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 97)( 90, 98)( 91, 99)( 92,100)
(101,105)(102,106)(103,107)(104,108)(109,117)(110,118)(111,119)(112,120)
(121,185)(122,186)(123,187)(124,188)(125,181)(126,182)(127,183)(128,184)
(129,197)(130,198)(131,199)(132,200)(133,193)(134,194)(135,195)(136,196)
(137,189)(138,190)(139,191)(140,192)(141,205)(142,206)(143,207)(144,208)
(145,201)(146,202)(147,203)(148,204)(149,217)(150,218)(151,219)(152,220)
(153,213)(154,214)(155,215)(156,216)(157,209)(158,210)(159,211)(160,212)
(161,225)(162,226)(163,227)(164,228)(165,221)(166,222)(167,223)(168,224)
(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)
(177,229)(178,230)(179,231)(180,232);;
s2 := ( 1,121)( 2,122)( 3,124)( 4,123)( 5,125)( 6,126)( 7,128)( 8,127)
( 9,129)( 10,130)( 11,132)( 12,131)( 13,133)( 14,134)( 15,136)( 16,135)
( 17,137)( 18,138)( 19,140)( 20,139)( 21,161)( 22,162)( 23,164)( 24,163)
( 25,165)( 26,166)( 27,168)( 28,167)( 29,169)( 30,170)( 31,172)( 32,171)
( 33,173)( 34,174)( 35,176)( 36,175)( 37,177)( 38,178)( 39,180)( 40,179)
( 41,141)( 42,142)( 43,144)( 44,143)( 45,145)( 46,146)( 47,148)( 48,147)
( 49,149)( 50,150)( 51,152)( 52,151)( 53,153)( 54,154)( 55,156)( 56,155)
( 57,157)( 58,158)( 59,160)( 60,159)( 61,181)( 62,182)( 63,184)( 64,183)
( 65,185)( 66,186)( 67,188)( 68,187)( 69,189)( 70,190)( 71,192)( 72,191)
( 73,193)( 74,194)( 75,196)( 76,195)( 77,197)( 78,198)( 79,200)( 80,199)
( 81,221)( 82,222)( 83,224)( 84,223)( 85,225)( 86,226)( 87,228)( 88,227)
( 89,229)( 90,230)( 91,232)( 92,231)( 93,233)( 94,234)( 95,236)( 96,235)
( 97,237)( 98,238)( 99,240)(100,239)(101,201)(102,202)(103,204)(104,203)
(105,205)(106,206)(107,208)(108,207)(109,209)(110,210)(111,212)(112,211)
(113,213)(114,214)(115,216)(116,215)(117,217)(118,218)(119,220)(120,219);;
s3 := ( 1, 21)( 2, 24)( 3, 23)( 4, 22)( 5, 25)( 6, 28)( 7, 27)( 8, 26)
( 9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)( 16, 34)
( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)( 54, 56)
( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)( 67, 87)
( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)( 75, 95)
( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)(110,112)
(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)(126,148)
(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)(134,156)
(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)(166,168)
(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)(185,205)
(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)(193,213)
(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)(222,224)
(226,228)(230,232)(234,236)(238,240);;
s4 := ( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 42)( 22, 41)( 23, 43)
( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 50)( 30, 49)( 31, 51)
( 32, 52)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 39, 59)
( 40, 60)( 61, 62)( 65, 66)( 69, 70)( 73, 74)( 77, 78)( 81,102)( 82,101)
( 83,103)( 84,104)( 85,106)( 86,105)( 87,107)( 88,108)( 89,110)( 90,109)
( 91,111)( 92,112)( 93,114)( 94,113)( 95,115)( 96,116)( 97,118)( 98,117)
( 99,119)(100,120)(121,122)(125,126)(129,130)(133,134)(137,138)(141,162)
(142,161)(143,163)(144,164)(145,166)(146,165)(147,167)(148,168)(149,170)
(150,169)(151,171)(152,172)(153,174)(154,173)(155,175)(156,176)(157,178)
(158,177)(159,179)(160,180)(181,182)(185,186)(189,190)(193,194)(197,198)
(201,222)(202,221)(203,223)(204,224)(205,226)(206,225)(207,227)(208,228)
(209,230)(210,229)(211,231)(212,232)(213,234)(214,233)(215,235)(216,236)
(217,238)(218,237)(219,239)(220,240);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)!( 5, 17)( 6, 18)( 7, 19)( 8, 20)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 25, 37)( 26, 38)( 27, 39)( 28, 40)( 29, 33)( 30, 34)( 31, 35)
( 32, 36)( 45, 57)( 46, 58)( 47, 59)( 48, 60)( 49, 53)( 50, 54)( 51, 55)
( 52, 56)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)
( 72, 76)( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)
(112,116)(125,137)(126,138)(127,139)(128,140)(129,133)(130,134)(131,135)
(132,136)(145,157)(146,158)(147,159)(148,160)(149,153)(150,154)(151,155)
(152,156)(165,177)(166,178)(167,179)(168,180)(169,173)(170,174)(171,175)
(172,176)(185,197)(186,198)(187,199)(188,200)(189,193)(190,194)(191,195)
(192,196)(205,217)(206,218)(207,219)(208,220)(209,213)(210,214)(211,215)
(212,216)(225,237)(226,238)(227,239)(228,240)(229,233)(230,234)(231,235)
(232,236);
s1 := Sym(240)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 17)( 10, 18)( 11, 19)
( 12, 20)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 29, 37)( 30, 38)( 31, 39)
( 32, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 57)( 50, 58)( 51, 59)
( 52, 60)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 97)( 90, 98)( 91, 99)
( 92,100)(101,105)(102,106)(103,107)(104,108)(109,117)(110,118)(111,119)
(112,120)(121,185)(122,186)(123,187)(124,188)(125,181)(126,182)(127,183)
(128,184)(129,197)(130,198)(131,199)(132,200)(133,193)(134,194)(135,195)
(136,196)(137,189)(138,190)(139,191)(140,192)(141,205)(142,206)(143,207)
(144,208)(145,201)(146,202)(147,203)(148,204)(149,217)(150,218)(151,219)
(152,220)(153,213)(154,214)(155,215)(156,216)(157,209)(158,210)(159,211)
(160,212)(161,225)(162,226)(163,227)(164,228)(165,221)(166,222)(167,223)
(168,224)(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)
(176,236)(177,229)(178,230)(179,231)(180,232);
s2 := Sym(240)!( 1,121)( 2,122)( 3,124)( 4,123)( 5,125)( 6,126)( 7,128)
( 8,127)( 9,129)( 10,130)( 11,132)( 12,131)( 13,133)( 14,134)( 15,136)
( 16,135)( 17,137)( 18,138)( 19,140)( 20,139)( 21,161)( 22,162)( 23,164)
( 24,163)( 25,165)( 26,166)( 27,168)( 28,167)( 29,169)( 30,170)( 31,172)
( 32,171)( 33,173)( 34,174)( 35,176)( 36,175)( 37,177)( 38,178)( 39,180)
( 40,179)( 41,141)( 42,142)( 43,144)( 44,143)( 45,145)( 46,146)( 47,148)
( 48,147)( 49,149)( 50,150)( 51,152)( 52,151)( 53,153)( 54,154)( 55,156)
( 56,155)( 57,157)( 58,158)( 59,160)( 60,159)( 61,181)( 62,182)( 63,184)
( 64,183)( 65,185)( 66,186)( 67,188)( 68,187)( 69,189)( 70,190)( 71,192)
( 72,191)( 73,193)( 74,194)( 75,196)( 76,195)( 77,197)( 78,198)( 79,200)
( 80,199)( 81,221)( 82,222)( 83,224)( 84,223)( 85,225)( 86,226)( 87,228)
( 88,227)( 89,229)( 90,230)( 91,232)( 92,231)( 93,233)( 94,234)( 95,236)
( 96,235)( 97,237)( 98,238)( 99,240)(100,239)(101,201)(102,202)(103,204)
(104,203)(105,205)(106,206)(107,208)(108,207)(109,209)(110,210)(111,212)
(112,211)(113,213)(114,214)(115,216)(116,215)(117,217)(118,218)(119,220)
(120,219);
s3 := Sym(240)!( 1, 21)( 2, 24)( 3, 23)( 4, 22)( 5, 25)( 6, 28)( 7, 27)
( 8, 26)( 9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)
( 54, 56)( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)
( 67, 87)( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)
( 75, 95)( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)
(110,112)(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)
(126,148)(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)
(134,156)(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)
(166,168)(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)
(185,205)(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)
(193,213)(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)
(222,224)(226,228)(230,232)(234,236)(238,240);
s4 := Sym(240)!( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 42)( 22, 41)
( 23, 43)( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 50)( 30, 49)
( 31, 51)( 32, 52)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 37, 58)( 38, 57)
( 39, 59)( 40, 60)( 61, 62)( 65, 66)( 69, 70)( 73, 74)( 77, 78)( 81,102)
( 82,101)( 83,103)( 84,104)( 85,106)( 86,105)( 87,107)( 88,108)( 89,110)
( 90,109)( 91,111)( 92,112)( 93,114)( 94,113)( 95,115)( 96,116)( 97,118)
( 98,117)( 99,119)(100,120)(121,122)(125,126)(129,130)(133,134)(137,138)
(141,162)(142,161)(143,163)(144,164)(145,166)(146,165)(147,167)(148,168)
(149,170)(150,169)(151,171)(152,172)(153,174)(154,173)(155,175)(156,176)
(157,178)(158,177)(159,179)(160,180)(181,182)(185,186)(189,190)(193,194)
(197,198)(201,222)(202,221)(203,223)(204,224)(205,226)(206,225)(207,227)
(208,228)(209,230)(210,229)(211,231)(212,232)(213,234)(214,233)(215,235)
(216,236)(217,238)(218,237)(219,239)(220,240);
poly := sub<Sym(240)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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