Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,12,4}

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

Overview

Group
SmallGroup(1920,240151)
Rank
4
Schläfli Type
{10,12,4}
Vertices, edges, …
10, 120, 48, 8
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

20-fold

24-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*s3*s2*s1*(s2*s3)^2> of order 2

4 facets

10 vertex figures

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

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