Polytope of Type {10,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,12,4}*960b
if this polytope has a name.
Group : SmallGroup(960,10951)
Rank : 4
Schlafli Type : {10,12,4}
Number of vertices, edges, etc : 10, 60, 24, 4
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,12,4,2} of size 1920
Vertex Figure Of :
{2,10,12,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,6,4}*480b
5-fold quotients : {2,12,4}*192b
10-fold quotients : {2,6,4}*96c
20-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,24,4}*1920c, {10,24,4}*1920d, {20,12,4}*1920b, {10,12,4}*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)(121,185)(122,187)(123,186)(124,188)(125,181)(126,183)
(127,182)(128,184)(129,197)(130,199)(131,198)(132,200)(133,193)(134,195)
(135,194)(136,196)(137,189)(138,191)(139,190)(140,192)(141,225)(142,227)
(143,226)(144,228)(145,221)(146,223)(147,222)(148,224)(149,237)(150,239)
(151,238)(152,240)(153,233)(154,235)(155,234)(156,236)(157,229)(158,231)
(159,230)(160,232)(161,205)(162,207)(163,206)(164,208)(165,201)(166,203)
(167,202)(168,204)(169,217)(170,219)(171,218)(172,220)(173,213)(174,215)
(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);;
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, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239);;
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, s3*s2*s1*s3*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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)(121,185)(122,187)(123,186)(124,188)(125,181)
(126,183)(127,182)(128,184)(129,197)(130,199)(131,198)(132,200)(133,193)
(134,195)(135,194)(136,196)(137,189)(138,191)(139,190)(140,192)(141,225)
(142,227)(143,226)(144,228)(145,221)(146,223)(147,222)(148,224)(149,237)
(150,239)(151,238)(152,240)(153,233)(154,235)(155,234)(156,236)(157,229)
(158,231)(159,230)(160,232)(161,205)(162,207)(163,206)(164,208)(165,201)
(166,203)(167,202)(168,204)(169,217)(170,219)(171,218)(172,220)(173,213)
(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);
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, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239);
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,
s3*s2*s1*s3*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope