Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,30}

Atlas Canonical Name {4,4,30}*1920a

Overview

Group
SmallGroup(1920,151293)
Rank
4
Schläfli Type
{4,4,30}
Vertices, edges, …
8, 16, 120, 30
Order of s0s1s2s3
60
Order of s0s1s2s3s2s1
2
Also known as
{{4,4}4,{4,30|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

5-fold

6-fold

8-fold

10-fold

12-fold

15-fold

16-fold

20-fold

24-fold

30-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)^2> of order 2

30 facets

  • 30 of 2-fold non-regular quotient of {4,4}*64

6 vertex figures

P/N, where N=<(s0*s1)^2> of order 2

30 facets

  • 30 of 2-fold non-regular quotient of {4,4}*64

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 51)( 37, 52)( 38, 53)( 39, 54)( 40, 55)( 41, 56)( 42, 57)( 43, 58)( 44, 59)( 45, 60)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(211,226)(212,227)(213,228)(214,229)(215,230)(216,231)(217,232)(218,233)(219,234)(220,235)(221,236)(222,237)(223,238)(224,239)(225,240);;
s1 := (121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240);;
s2 := (  1,181)(  2,185)(  3,184)(  4,183)(  5,182)(  6,191)(  7,195)(  8,194)(  9,193)( 10,192)( 11,186)( 12,190)( 13,189)( 14,188)( 15,187)( 16,196)( 17,200)( 18,199)( 19,198)( 20,197)( 21,206)( 22,210)( 23,209)( 24,208)( 25,207)( 26,201)( 27,205)( 28,204)( 29,203)( 30,202)( 31,211)( 32,215)( 33,214)( 34,213)( 35,212)( 36,221)( 37,225)( 38,224)( 39,223)( 40,222)( 41,216)( 42,220)( 43,219)( 44,218)( 45,217)( 46,226)( 47,230)( 48,229)( 49,228)( 50,227)( 51,236)( 52,240)( 53,239)( 54,238)( 55,237)( 56,231)( 57,235)( 58,234)( 59,233)( 60,232)( 61,121)( 62,125)( 63,124)( 64,123)( 65,122)( 66,131)( 67,135)( 68,134)( 69,133)( 70,132)( 71,126)( 72,130)( 73,129)( 74,128)( 75,127)( 76,136)( 77,140)( 78,139)( 79,138)( 80,137)( 81,146)( 82,150)( 83,149)( 84,148)( 85,147)( 86,141)( 87,145)( 88,144)( 89,143)( 90,142)( 91,151)( 92,155)( 93,154)( 94,153)( 95,152)( 96,161)( 97,165)( 98,164)( 99,163)(100,162)(101,156)(102,160)(103,159)(104,158)(105,157)(106,166)(107,170)(108,169)(109,168)(110,167)(111,176)(112,180)(113,179)(114,178)(115,177)(116,171)(117,175)(118,174)(119,173)(120,172);;
s3 := (  1, 67)(  2, 66)(  3, 70)(  4, 69)(  5, 68)(  6, 62)(  7, 61)(  8, 65)(  9, 64)( 10, 63)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 77)( 22, 76)( 23, 80)( 24, 79)( 25, 78)( 26, 87)( 27, 86)( 28, 90)( 29, 89)( 30, 88)( 31, 97)( 32, 96)( 33,100)( 34, 99)( 35, 98)( 36, 92)( 37, 91)( 38, 95)( 39, 94)( 40, 93)( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 46,112)( 47,111)( 48,115)( 49,114)( 50,113)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,117)( 57,116)( 58,120)( 59,119)( 60,118)(121,187)(122,186)(123,190)(124,189)(125,188)(126,182)(127,181)(128,185)(129,184)(130,183)(131,192)(132,191)(133,195)(134,194)(135,193)(136,202)(137,201)(138,205)(139,204)(140,203)(141,197)(142,196)(143,200)(144,199)(145,198)(146,207)(147,206)(148,210)(149,209)(150,208)(151,217)(152,216)(153,220)(154,219)(155,218)(156,212)(157,211)(158,215)(159,214)(160,213)(161,222)(162,221)(163,225)(164,224)(165,223)(166,232)(167,231)(168,235)(169,234)(170,233)(171,227)(172,226)(173,230)(174,229)(175,228)(176,237)(177,236)(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*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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)!( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 51)( 37, 52)( 38, 53)( 39, 54)( 40, 55)( 41, 56)( 42, 57)( 43, 58)( 44, 59)( 45, 60)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)(211,226)(212,227)(213,228)(214,229)(215,230)(216,231)(217,232)(218,233)(219,234)(220,235)(221,236)(222,237)(223,238)(224,239)(225,240);
s1 := Sym(240)!(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240);
s2 := Sym(240)!(  1,181)(  2,185)(  3,184)(  4,183)(  5,182)(  6,191)(  7,195)(  8,194)(  9,193)( 10,192)( 11,186)( 12,190)( 13,189)( 14,188)( 15,187)( 16,196)( 17,200)( 18,199)( 19,198)( 20,197)( 21,206)( 22,210)( 23,209)( 24,208)( 25,207)( 26,201)( 27,205)( 28,204)( 29,203)( 30,202)( 31,211)( 32,215)( 33,214)( 34,213)( 35,212)( 36,221)( 37,225)( 38,224)( 39,223)( 40,222)( 41,216)( 42,220)( 43,219)( 44,218)( 45,217)( 46,226)( 47,230)( 48,229)( 49,228)( 50,227)( 51,236)( 52,240)( 53,239)( 54,238)( 55,237)( 56,231)( 57,235)( 58,234)( 59,233)( 60,232)( 61,121)( 62,125)( 63,124)( 64,123)( 65,122)( 66,131)( 67,135)( 68,134)( 69,133)( 70,132)( 71,126)( 72,130)( 73,129)( 74,128)( 75,127)( 76,136)( 77,140)( 78,139)( 79,138)( 80,137)( 81,146)( 82,150)( 83,149)( 84,148)( 85,147)( 86,141)( 87,145)( 88,144)( 89,143)( 90,142)( 91,151)( 92,155)( 93,154)( 94,153)( 95,152)( 96,161)( 97,165)( 98,164)( 99,163)(100,162)(101,156)(102,160)(103,159)(104,158)(105,157)(106,166)(107,170)(108,169)(109,168)(110,167)(111,176)(112,180)(113,179)(114,178)(115,177)(116,171)(117,175)(118,174)(119,173)(120,172);
s3 := Sym(240)!(  1, 67)(  2, 66)(  3, 70)(  4, 69)(  5, 68)(  6, 62)(  7, 61)(  8, 65)(  9, 64)( 10, 63)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 77)( 22, 76)( 23, 80)( 24, 79)( 25, 78)( 26, 87)( 27, 86)( 28, 90)( 29, 89)( 30, 88)( 31, 97)( 32, 96)( 33,100)( 34, 99)( 35, 98)( 36, 92)( 37, 91)( 38, 95)( 39, 94)( 40, 93)( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 46,112)( 47,111)( 48,115)( 49,114)( 50,113)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,117)( 57,116)( 58,120)( 59,119)( 60,118)(121,187)(122,186)(123,190)(124,189)(125,188)(126,182)(127,181)(128,185)(129,184)(130,183)(131,192)(132,191)(133,195)(134,194)(135,193)(136,202)(137,201)(138,205)(139,204)(140,203)(141,197)(142,196)(143,200)(144,199)(145,198)(146,207)(147,206)(148,210)(149,209)(150,208)(151,217)(152,216)(153,220)(154,219)(155,218)(156,212)(157,211)(158,215)(159,214)(160,213)(161,222)(162,221)(163,225)(164,224)(165,223)(166,232)(167,231)(168,235)(169,234)(170,233)(171,227)(172,226)(173,230)(174,229)(175,228)(176,237)(177,236)(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*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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.