Polytope of Type {4,10,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,10,12}*960
Also Known As : {{4,10|2},{10,12|2}}. if this polytope has another name.
Group : SmallGroup(960,7449)
Rank : 4
Schlafli Type : {4,10,12}
Number of vertices, edges, etc : 4, 20, 60, 12
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,10,12,2} of size 1920
Vertex Figure Of :
{2,4,10,12} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,10,12}*480, {4,10,6}*480
3-fold quotients : {4,10,4}*320
4-fold quotients : {2,10,6}*240
5-fold quotients : {4,2,12}*192
6-fold quotients : {2,10,4}*160, {4,10,2}*160
10-fold quotients : {2,2,12}*96, {4,2,6}*96
12-fold quotients : {2,10,2}*80
15-fold quotients : {4,2,4}*64
20-fold quotients : {4,2,3}*48, {2,2,6}*48
24-fold quotients : {2,5,2}*40
30-fold quotients : {2,2,4}*32, {4,2,2}*32
40-fold quotients : {2,2,3}*24
60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,20,12}*1920, {8,10,12}*1920, {4,10,24}*1920
Permutation Representation (GAP) :
s0 := ( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)
( 9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)
( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)
( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)
( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)( 40,160)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,181)( 77,182)( 78,183)( 79,184)( 80,185)
( 81,186)( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)( 88,193)
( 89,194)( 90,195)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)( 96,231)
( 97,232)( 98,233)( 99,234)(100,235)(101,236)(102,237)(103,238)(104,239)
(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)
(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225);;
s1 := ( 2, 5)( 3, 4)( 7, 10)( 8, 9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99)
(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)
(121,136)(122,140)(123,139)(124,138)(125,137)(126,141)(127,145)(128,144)
(129,143)(130,142)(131,146)(132,150)(133,149)(134,148)(135,147)(151,166)
(152,170)(153,169)(154,168)(155,167)(156,171)(157,175)(158,174)(159,173)
(160,172)(161,176)(162,180)(163,179)(164,178)(165,177)(181,196)(182,200)
(183,199)(184,198)(185,197)(186,201)(187,205)(188,204)(189,203)(190,202)
(191,206)(192,210)(193,209)(194,208)(195,207)(211,226)(212,230)(213,229)
(214,228)(215,227)(216,231)(217,235)(218,234)(219,233)(220,232)(221,236)
(222,240)(223,239)(224,238)(225,237);;
s2 := ( 1, 62)( 2, 61)( 3, 65)( 4, 64)( 5, 63)( 6, 72)( 7, 71)( 8, 75)
( 9, 74)( 10, 73)( 11, 67)( 12, 66)( 13, 70)( 14, 69)( 15, 68)( 16, 77)
( 17, 76)( 18, 80)( 19, 79)( 20, 78)( 21, 87)( 22, 86)( 23, 90)( 24, 89)
( 25, 88)( 26, 82)( 27, 81)( 28, 85)( 29, 84)( 30, 83)( 31, 92)( 32, 91)
( 33, 95)( 34, 94)( 35, 93)( 36,102)( 37,101)( 38,105)( 39,104)( 40,103)
( 41, 97)( 42, 96)( 43,100)( 44, 99)( 45, 98)( 46,107)( 47,106)( 48,110)
( 49,109)( 50,108)( 51,117)( 52,116)( 53,120)( 54,119)( 55,118)( 56,112)
( 57,111)( 58,115)( 59,114)( 60,113)(121,197)(122,196)(123,200)(124,199)
(125,198)(126,207)(127,206)(128,210)(129,209)(130,208)(131,202)(132,201)
(133,205)(134,204)(135,203)(136,182)(137,181)(138,185)(139,184)(140,183)
(141,192)(142,191)(143,195)(144,194)(145,193)(146,187)(147,186)(148,190)
(149,189)(150,188)(151,227)(152,226)(153,230)(154,229)(155,228)(156,237)
(157,236)(158,240)(159,239)(160,238)(161,232)(162,231)(163,235)(164,234)
(165,233)(166,212)(167,211)(168,215)(169,214)(170,213)(171,222)(172,221)
(173,225)(174,224)(175,223)(176,217)(177,216)(178,220)(179,219)(180,218);;
s3 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 21)( 17, 22)( 18, 23)
( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 51)
( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 96)( 62, 97)( 63, 98)( 64, 99)
( 65,100)( 66, 91)( 67, 92)( 68, 93)( 69, 94)( 70, 95)( 71,101)( 72,102)
( 73,103)( 74,104)( 75,105)( 76,111)( 77,112)( 78,113)( 79,114)( 80,115)
( 81,106)( 82,107)( 83,108)( 84,109)( 85,110)( 86,116)( 87,117)( 88,118)
( 89,119)( 90,120)(121,126)(122,127)(123,128)(124,129)(125,130)(136,141)
(137,142)(138,143)(139,144)(140,145)(151,156)(152,157)(153,158)(154,159)
(155,160)(166,171)(167,172)(168,173)(169,174)(170,175)(181,216)(182,217)
(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)(190,215)
(191,221)(192,222)(193,223)(194,224)(195,225)(196,231)(197,232)(198,233)
(199,234)(200,235)(201,226)(202,227)(203,228)(204,229)(205,230)(206,236)
(207,237)(208,238)(209,239)(210,240);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(240)!( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)
( 8,128)( 9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)
( 16,136)( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)
( 24,144)( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)
( 32,152)( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)
( 40,160)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,181)( 77,182)( 78,183)( 79,184)
( 80,185)( 81,186)( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)
( 88,193)( 89,194)( 90,195)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)
( 96,231)( 97,232)( 98,233)( 99,234)(100,235)(101,236)(102,237)(103,238)
(104,239)(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)
(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)
(120,225);
s1 := Sym(240)!( 2, 5)( 3, 4)( 7, 10)( 8, 9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)
( 78, 79)( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)
( 98, 99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)
(118,119)(121,136)(122,140)(123,139)(124,138)(125,137)(126,141)(127,145)
(128,144)(129,143)(130,142)(131,146)(132,150)(133,149)(134,148)(135,147)
(151,166)(152,170)(153,169)(154,168)(155,167)(156,171)(157,175)(158,174)
(159,173)(160,172)(161,176)(162,180)(163,179)(164,178)(165,177)(181,196)
(182,200)(183,199)(184,198)(185,197)(186,201)(187,205)(188,204)(189,203)
(190,202)(191,206)(192,210)(193,209)(194,208)(195,207)(211,226)(212,230)
(213,229)(214,228)(215,227)(216,231)(217,235)(218,234)(219,233)(220,232)
(221,236)(222,240)(223,239)(224,238)(225,237);
s2 := Sym(240)!( 1, 62)( 2, 61)( 3, 65)( 4, 64)( 5, 63)( 6, 72)( 7, 71)
( 8, 75)( 9, 74)( 10, 73)( 11, 67)( 12, 66)( 13, 70)( 14, 69)( 15, 68)
( 16, 77)( 17, 76)( 18, 80)( 19, 79)( 20, 78)( 21, 87)( 22, 86)( 23, 90)
( 24, 89)( 25, 88)( 26, 82)( 27, 81)( 28, 85)( 29, 84)( 30, 83)( 31, 92)
( 32, 91)( 33, 95)( 34, 94)( 35, 93)( 36,102)( 37,101)( 38,105)( 39,104)
( 40,103)( 41, 97)( 42, 96)( 43,100)( 44, 99)( 45, 98)( 46,107)( 47,106)
( 48,110)( 49,109)( 50,108)( 51,117)( 52,116)( 53,120)( 54,119)( 55,118)
( 56,112)( 57,111)( 58,115)( 59,114)( 60,113)(121,197)(122,196)(123,200)
(124,199)(125,198)(126,207)(127,206)(128,210)(129,209)(130,208)(131,202)
(132,201)(133,205)(134,204)(135,203)(136,182)(137,181)(138,185)(139,184)
(140,183)(141,192)(142,191)(143,195)(144,194)(145,193)(146,187)(147,186)
(148,190)(149,189)(150,188)(151,227)(152,226)(153,230)(154,229)(155,228)
(156,237)(157,236)(158,240)(159,239)(160,238)(161,232)(162,231)(163,235)
(164,234)(165,233)(166,212)(167,211)(168,215)(169,214)(170,213)(171,222)
(172,221)(173,225)(174,224)(175,223)(176,217)(177,216)(178,220)(179,219)
(180,218);
s3 := Sym(240)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 21)( 17, 22)
( 18, 23)( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)
( 46, 51)( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 96)( 62, 97)( 63, 98)
( 64, 99)( 65,100)( 66, 91)( 67, 92)( 68, 93)( 69, 94)( 70, 95)( 71,101)
( 72,102)( 73,103)( 74,104)( 75,105)( 76,111)( 77,112)( 78,113)( 79,114)
( 80,115)( 81,106)( 82,107)( 83,108)( 84,109)( 85,110)( 86,116)( 87,117)
( 88,118)( 89,119)( 90,120)(121,126)(122,127)(123,128)(124,129)(125,130)
(136,141)(137,142)(138,143)(139,144)(140,145)(151,156)(152,157)(153,158)
(154,159)(155,160)(166,171)(167,172)(168,173)(169,174)(170,175)(181,216)
(182,217)(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)
(190,215)(191,221)(192,222)(193,223)(194,224)(195,225)(196,231)(197,232)
(198,233)(199,234)(200,235)(201,226)(202,227)(203,228)(204,229)(205,230)
(206,236)(207,237)(208,238)(209,239)(210,240);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*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 >;
References : None.
to this polytope