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