Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,6,4}

Atlas Canonical Name {10,6,4}*1920b

Overview

Group
SmallGroup(1920,240151)
Rank
4
Schläfli Type
{10,6,4}
Vertices, edges, …
10, 120, 48, 16
Order of s0s1s2s3
60
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

5-fold

8-fold

10-fold

12-fold

20-fold

24-fold

40-fold

48-fold

60-fold

80-fold

120-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s2*s3*s2*s1*(s2*s3)^2> of order 2

8 facets

10 vertex figures

P/N, where N=<(s1*s2*s3*s2)^2> of order 2

8 facets

10 vertex figures

P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2> of order 2

8 facets

10 vertex figures

P/N, where N=<(s1*s2)^2> of order 3

8 facets

10 vertex figures

P/N, where N=<(s2*s3)^2, s1*s2*s3*s2*s1*s3> of order 4

4 facets

10 vertex figures

Representations

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