Polytope of Type {6,6,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,10}*960
if this polytope has a name.
Group : SmallGroup(960,11372)
Rank : 4
Schlafli Type : {6,6,10}
Number of vertices, edges, etc : 8, 24, 40, 10
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,10,2} of size 1920
Vertex Figure Of :
{2,6,6,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,10}*480
5-fold quotients : {6,6,2}*192
10-fold quotients : {3,6,2}*96, {6,3,2}*96
12-fold quotients : {2,2,10}*80
20-fold quotients : {3,3,2}*48
24-fold quotients : {2,2,5}*40
60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6,20}*1920, {6,12,10}*1920a, {12,6,10}*1920a, {6,12,10}*1920b, {12,6,10}*1920b, {6,6,10}*1920
Irregular Quotients (of which this is a minimal cover):
None.
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)( 5, 20)( 6, 18)( 7, 19)( 8, 17)( 9, 16)( 10, 14)( 11, 15)( 12, 13)( 21, 44)( 22, 42)( 23, 43)( 24, 41)( 25, 60)( 26, 58)( 27, 59)( 28, 57)( 29, 56)( 30, 54)( 31, 55)( 32, 53)( 33, 52)( 34, 50)( 35, 51)( 36, 49)( 37, 48)( 38, 46)( 39, 47)( 40, 45)( 61, 64)( 65, 80)( 66, 78)( 67, 79)( 68, 77)( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 81,104)( 82,102)( 83,103)( 84,101)( 85,120)( 86,118)( 87,119)( 88,117)( 89,116)( 90,114)( 91,115)( 92,113)( 93,112)( 94,110)( 95,111)( 96,109)( 97,108)( 98,106)( 99,107)(100,105)(121,124)(125,140)(126,138)(127,139)(128,137)(129,136)(130,134)(131,135)(132,133)(141,164)(142,162)(143,163)(144,161)(145,180)(146,178)(147,179)(148,177)(149,176)(150,174)(151,175)(152,173)(153,172)(154,170)(155,171)(156,169)(157,168)(158,166)(159,167)(160,165)(181,184)(185,200)(186,198)(187,199)(188,197)(189,196)(190,194)(191,195)(192,193)(201,224)(202,222)(203,223)(204,221)(205,240)(206,238)(207,239)(208,237)(209,236)(210,234)(211,235)(212,233)(213,232)(214,230)(215,231)(216,229)(217,228)(218,226)(219,227)(220,225);;
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*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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)( 5, 20)( 6, 18)( 7, 19)( 8, 17)( 9, 16)( 10, 14)( 11, 15)( 12, 13)( 21, 44)( 22, 42)( 23, 43)( 24, 41)( 25, 60)( 26, 58)( 27, 59)( 28, 57)( 29, 56)( 30, 54)( 31, 55)( 32, 53)( 33, 52)( 34, 50)( 35, 51)( 36, 49)( 37, 48)( 38, 46)( 39, 47)( 40, 45)( 61, 64)( 65, 80)( 66, 78)( 67, 79)( 68, 77)( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 81,104)( 82,102)( 83,103)( 84,101)( 85,120)( 86,118)( 87,119)( 88,117)( 89,116)( 90,114)( 91,115)( 92,113)( 93,112)( 94,110)( 95,111)( 96,109)( 97,108)( 98,106)( 99,107)(100,105)(121,124)(125,140)(126,138)(127,139)(128,137)(129,136)(130,134)(131,135)(132,133)(141,164)(142,162)(143,163)(144,161)(145,180)(146,178)(147,179)(148,177)(149,176)(150,174)(151,175)(152,173)(153,172)(154,170)(155,171)(156,169)(157,168)(158,166)(159,167)(160,165)(181,184)(185,200)(186,198)(187,199)(188,197)(189,196)(190,194)(191,195)(192,193)(201,224)(202,222)(203,223)(204,221)(205,240)(206,238)(207,239)(208,237)(209,236)(210,234)(211,235)(212,233)(213,232)(214,230)(215,231)(216,229)(217,228)(218,226)(219,227)(220,225);
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*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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