Polytope of Type {10,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6,4}*960e
if this polytope has a name.
Group : SmallGroup(960,11372)
Rank : 4
Schlafli Type : {10,6,4}
Number of vertices, edges, etc : 10, 60, 24, 8
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,6,4,2} of size 1920
Vertex Figure Of :
{2,10,6,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,6,4}*480b
4-fold quotients : {10,6,2}*240
5-fold quotients : {2,6,4}*192
10-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
12-fold quotients : {10,2,2}*80
20-fold quotients : {2,3,4}*48, {2,6,2}*48
24-fold quotients : {5,2,2}*40
40-fold quotients : {2,3,2}*24
60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,12,4}*1920b, {20,6,4}*1920a, {10,6,4}*1920b, {10,12,4}*1920c, {10,6,8}*1920a, {10,6,8}*1920b
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, 7)( 3, 6)( 4, 8)( 9, 17)( 10, 19)( 11, 18)( 12, 20)
( 14, 15)( 21, 45)( 22, 47)( 23, 46)( 24, 48)( 25, 41)( 26, 43)( 27, 42)
( 28, 44)( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 53)( 34, 55)( 35, 54)
( 36, 56)( 37, 49)( 38, 51)( 39, 50)( 40, 52)( 61, 65)( 62, 67)( 63, 66)
( 64, 68)( 69, 77)( 70, 79)( 71, 78)( 72, 80)( 74, 75)( 81,105)( 82,107)
( 83,106)( 84,108)( 85,101)( 86,103)( 87,102)( 88,104)( 89,117)( 90,119)
( 91,118)( 92,120)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)( 98,111)
( 99,110)(100,112)(121,125)(122,127)(123,126)(124,128)(129,137)(130,139)
(131,138)(132,140)(134,135)(141,165)(142,167)(143,166)(144,168)(145,161)
(146,163)(147,162)(148,164)(149,177)(150,179)(151,178)(152,180)(153,173)
(154,175)(155,174)(156,176)(157,169)(158,171)(159,170)(160,172)(181,185)
(182,187)(183,186)(184,188)(189,197)(190,199)(191,198)(192,200)(194,195)
(201,225)(202,227)(203,226)(204,228)(205,221)(206,223)(207,222)(208,224)
(209,237)(210,239)(211,238)(212,240)(213,233)(214,235)(215,234)(216,236)
(217,229)(218,231)(219,230)(220,232);;
s2 := ( 1,141)( 2,142)( 3,144)( 4,143)( 5,145)( 6,146)( 7,148)( 8,147)
( 9,149)( 10,150)( 11,152)( 12,151)( 13,153)( 14,154)( 15,156)( 16,155)
( 17,157)( 18,158)( 19,160)( 20,159)( 21,121)( 22,122)( 23,124)( 24,123)
( 25,125)( 26,126)( 27,128)( 28,127)( 29,129)( 30,130)( 31,132)( 32,131)
( 33,133)( 34,134)( 35,136)( 36,135)( 37,137)( 38,138)( 39,140)( 40,139)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,169)( 50,170)( 51,172)( 52,171)( 53,173)( 54,174)( 55,176)( 56,175)
( 57,177)( 58,178)( 59,180)( 60,179)( 61,201)( 62,202)( 63,204)( 64,203)
( 65,205)( 66,206)( 67,208)( 68,207)( 69,209)( 70,210)( 71,212)( 72,211)
( 73,213)( 74,214)( 75,216)( 76,215)( 77,217)( 78,218)( 79,220)( 80,219)
( 81,181)( 82,182)( 83,184)( 84,183)( 85,185)( 86,186)( 87,188)( 88,187)
( 89,189)( 90,190)( 91,192)( 92,191)( 93,193)( 94,194)( 95,196)( 96,195)
( 97,197)( 98,198)( 99,200)(100,199)(101,221)(102,222)(103,224)(104,223)
(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)(112,231)
(113,233)(114,234)(115,236)(116,235)(117,237)(118,238)(119,240)(120,239);;
s3 := ( 1, 64)( 2, 63)( 3, 62)( 4, 61)( 5, 68)( 6, 67)( 7, 66)( 8, 65)
( 9, 72)( 10, 71)( 11, 70)( 12, 69)( 13, 76)( 14, 75)( 15, 74)( 16, 73)
( 17, 80)( 18, 79)( 19, 78)( 20, 77)( 21, 84)( 22, 83)( 23, 82)( 24, 81)
( 25, 88)( 26, 87)( 27, 86)( 28, 85)( 29, 92)( 30, 91)( 31, 90)( 32, 89)
( 33, 96)( 34, 95)( 35, 94)( 36, 93)( 37,100)( 38, 99)( 39, 98)( 40, 97)
( 41,104)( 42,103)( 43,102)( 44,101)( 45,108)( 46,107)( 47,106)( 48,105)
( 49,112)( 50,111)( 51,110)( 52,109)( 53,116)( 54,115)( 55,114)( 56,113)
( 57,120)( 58,119)( 59,118)( 60,117)(121,184)(122,183)(123,182)(124,181)
(125,188)(126,187)(127,186)(128,185)(129,192)(130,191)(131,190)(132,189)
(133,196)(134,195)(135,194)(136,193)(137,200)(138,199)(139,198)(140,197)
(141,204)(142,203)(143,202)(144,201)(145,208)(146,207)(147,206)(148,205)
(149,212)(150,211)(151,210)(152,209)(153,216)(154,215)(155,214)(156,213)
(157,220)(158,219)(159,218)(160,217)(161,224)(162,223)(163,222)(164,221)
(165,228)(166,227)(167,226)(168,225)(169,232)(170,231)(171,230)(172,229)
(173,236)(174,235)(175,234)(176,233)(177,240)(178,239)(179,238)(180,237);;
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*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
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, 7)( 3, 6)( 4, 8)( 9, 17)( 10, 19)( 11, 18)
( 12, 20)( 14, 15)( 21, 45)( 22, 47)( 23, 46)( 24, 48)( 25, 41)( 26, 43)
( 27, 42)( 28, 44)( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 53)( 34, 55)
( 35, 54)( 36, 56)( 37, 49)( 38, 51)( 39, 50)( 40, 52)( 61, 65)( 62, 67)
( 63, 66)( 64, 68)( 69, 77)( 70, 79)( 71, 78)( 72, 80)( 74, 75)( 81,105)
( 82,107)( 83,106)( 84,108)( 85,101)( 86,103)( 87,102)( 88,104)( 89,117)
( 90,119)( 91,118)( 92,120)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)
( 98,111)( 99,110)(100,112)(121,125)(122,127)(123,126)(124,128)(129,137)
(130,139)(131,138)(132,140)(134,135)(141,165)(142,167)(143,166)(144,168)
(145,161)(146,163)(147,162)(148,164)(149,177)(150,179)(151,178)(152,180)
(153,173)(154,175)(155,174)(156,176)(157,169)(158,171)(159,170)(160,172)
(181,185)(182,187)(183,186)(184,188)(189,197)(190,199)(191,198)(192,200)
(194,195)(201,225)(202,227)(203,226)(204,228)(205,221)(206,223)(207,222)
(208,224)(209,237)(210,239)(211,238)(212,240)(213,233)(214,235)(215,234)
(216,236)(217,229)(218,231)(219,230)(220,232);
s2 := Sym(240)!( 1,141)( 2,142)( 3,144)( 4,143)( 5,145)( 6,146)( 7,148)
( 8,147)( 9,149)( 10,150)( 11,152)( 12,151)( 13,153)( 14,154)( 15,156)
( 16,155)( 17,157)( 18,158)( 19,160)( 20,159)( 21,121)( 22,122)( 23,124)
( 24,123)( 25,125)( 26,126)( 27,128)( 28,127)( 29,129)( 30,130)( 31,132)
( 32,131)( 33,133)( 34,134)( 35,136)( 36,135)( 37,137)( 38,138)( 39,140)
( 40,139)( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)
( 48,167)( 49,169)( 50,170)( 51,172)( 52,171)( 53,173)( 54,174)( 55,176)
( 56,175)( 57,177)( 58,178)( 59,180)( 60,179)( 61,201)( 62,202)( 63,204)
( 64,203)( 65,205)( 66,206)( 67,208)( 68,207)( 69,209)( 70,210)( 71,212)
( 72,211)( 73,213)( 74,214)( 75,216)( 76,215)( 77,217)( 78,218)( 79,220)
( 80,219)( 81,181)( 82,182)( 83,184)( 84,183)( 85,185)( 86,186)( 87,188)
( 88,187)( 89,189)( 90,190)( 91,192)( 92,191)( 93,193)( 94,194)( 95,196)
( 96,195)( 97,197)( 98,198)( 99,200)(100,199)(101,221)(102,222)(103,224)
(104,223)(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)
(112,231)(113,233)(114,234)(115,236)(116,235)(117,237)(118,238)(119,240)
(120,239);
s3 := Sym(240)!( 1, 64)( 2, 63)( 3, 62)( 4, 61)( 5, 68)( 6, 67)( 7, 66)
( 8, 65)( 9, 72)( 10, 71)( 11, 70)( 12, 69)( 13, 76)( 14, 75)( 15, 74)
( 16, 73)( 17, 80)( 18, 79)( 19, 78)( 20, 77)( 21, 84)( 22, 83)( 23, 82)
( 24, 81)( 25, 88)( 26, 87)( 27, 86)( 28, 85)( 29, 92)( 30, 91)( 31, 90)
( 32, 89)( 33, 96)( 34, 95)( 35, 94)( 36, 93)( 37,100)( 38, 99)( 39, 98)
( 40, 97)( 41,104)( 42,103)( 43,102)( 44,101)( 45,108)( 46,107)( 47,106)
( 48,105)( 49,112)( 50,111)( 51,110)( 52,109)( 53,116)( 54,115)( 55,114)
( 56,113)( 57,120)( 58,119)( 59,118)( 60,117)(121,184)(122,183)(123,182)
(124,181)(125,188)(126,187)(127,186)(128,185)(129,192)(130,191)(131,190)
(132,189)(133,196)(134,195)(135,194)(136,193)(137,200)(138,199)(139,198)
(140,197)(141,204)(142,203)(143,202)(144,201)(145,208)(146,207)(147,206)
(148,205)(149,212)(150,211)(151,210)(152,209)(153,216)(154,215)(155,214)
(156,213)(157,220)(158,219)(159,218)(160,217)(161,224)(162,223)(163,222)
(164,221)(165,228)(166,227)(167,226)(168,225)(169,232)(170,231)(171,230)
(172,229)(173,236)(174,235)(175,234)(176,233)(177,240)(178,239)(179,238)
(180,237);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
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