Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,4,12}

Atlas Canonical Name {10,4,12}*1920c

Overview

Group
SmallGroup(1920,240151)
Rank
4
Schläfli Type
{10,4,12}
Vertices, edges, …
10, 40, 48, 24
Order of s0s1s2s3
30
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

16-fold

20-fold

24-fold

32-fold

40-fold

48-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)^2> of order 2

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