Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*1936

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Overview

Group
SmallGroup(1936,161)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
242, 484, 242
Order of s0s1s2
22
Order of s0s1s2s1
22
Also known as
{4,4}(11,11), {4,4}22. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

242-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=<(s0*s1*s2*s1)^10*s0*s1*s2> of order 2

121 facets

121 vertex figures

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

122 facets

122 vertex figures

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^10*s0*s1*s2> of order 4

61 facets

61 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2> of order 11

22 facets

22 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*(s1*s2)^2> of order 11

22 facets

22 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1*s0*s1)^2*(s2*s1*s0)^2*s2*s1*s2> of order 11

22 facets

22 vertex figures

P/N, where N=<(s0*s1)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 22

12 facets

12 vertex figures

P/N, where N=<(s0*s1)^2, (s1*s2*s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1*s2> of order 22

12 facets

12 vertex figures

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^2> of order 22

12 facets

12 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  2,112)(  3,102)(  4, 92)(  5, 82)(  6, 72)(  7, 62)(  8, 52)(  9, 42)( 10, 32)( 11, 22)( 12,111)( 13,101)( 14, 91)( 15, 81)( 16, 71)( 17, 61)( 18, 51)( 19, 41)( 20, 31)( 23,100)( 24, 90)( 25, 80)( 26, 70)( 27, 60)( 28, 50)( 29, 40)( 33,121)( 34, 89)( 35, 79)( 36, 69)( 37, 59)( 38, 49)( 43,120)( 44,110)( 45, 78)( 46, 68)( 47, 58)( 53,119)( 54,109)( 55, 99)( 56, 67)( 63,118)( 64,108)( 65, 98)( 66, 88)( 73,117)( 74,107)( 75, 97)( 76, 87)( 83,116)( 84,106)( 85, 96)( 93,115)( 94,105)(103,114)(123,233)(124,223)(125,213)(126,203)(127,193)(128,183)(129,173)(130,163)(131,153)(132,143)(133,232)(134,222)(135,212)(136,202)(137,192)(138,182)(139,172)(140,162)(141,152)(144,221)(145,211)(146,201)(147,191)(148,181)(149,171)(150,161)(154,242)(155,210)(156,200)(157,190)(158,180)(159,170)(164,241)(165,231)(166,199)(167,189)(168,179)(174,240)(175,230)(176,220)(177,188)(184,239)(185,229)(186,219)(187,209)(194,238)(195,228)(196,218)(197,208)(204,237)(205,227)(206,217)(214,236)(215,226)(224,235);;
s1 := ( 12,113)( 13,114)( 14,115)( 15,116)( 16,117)( 17,118)( 18,119)( 19,120)( 20,121)( 21,111)( 22,112)( 23,104)( 24,105)( 25,106)( 26,107)( 27,108)( 28,109)( 29,110)( 30,100)( 31,101)( 32,102)( 33,103)( 34, 95)( 35, 96)( 36, 97)( 37, 98)( 38, 99)( 39, 89)( 40, 90)( 41, 91)( 42, 92)( 43, 93)( 44, 94)( 45, 86)( 46, 87)( 47, 88)( 48, 78)( 49, 79)( 50, 80)( 51, 81)( 52, 82)( 53, 83)( 54, 84)( 55, 85)( 56, 77)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 61, 71)( 62, 72)( 63, 73)( 64, 74)( 65, 75)( 66, 76)(133,234)(134,235)(135,236)(136,237)(137,238)(138,239)(139,240)(140,241)(141,242)(142,232)(143,233)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,221)(152,222)(153,223)(154,224)(155,216)(156,217)(157,218)(158,219)(159,220)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,207)(167,208)(168,209)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(175,205)(176,206)(177,198)(178,188)(179,189)(180,190)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197);;
s2 := (  1,178)(  2,188)(  3,209)(  4,219)(  5,229)(  6,239)(  7,128)(  8,138)(  9,148)( 10,158)( 11,168)( 12,189)( 13,199)( 14,220)( 15,230)( 16,240)( 17,129)( 18,139)( 19,149)( 20,159)( 21,169)( 22,179)( 23,200)( 24,210)( 25,231)( 26,241)( 27,130)( 28,140)( 29,150)( 30,160)( 31,170)( 32,180)( 33,190)( 34,211)( 35,221)( 36,242)( 37,131)( 38,141)( 39,151)( 40,161)( 41,171)( 42,181)( 43,191)( 44,201)( 45,222)( 46,232)( 47,132)( 48,142)( 49,152)( 50,162)( 51,172)( 52,182)( 53,192)( 54,202)( 55,212)( 56,233)( 57,122)( 58,143)( 59,153)( 60,163)( 61,173)( 62,183)( 63,193)( 64,203)( 65,213)( 66,223)( 67,123)( 68,133)( 69,154)( 70,164)( 71,174)( 72,184)( 73,194)( 74,204)( 75,214)( 76,224)( 77,234)( 78,134)( 79,144)( 80,165)( 81,175)( 82,185)( 83,195)( 84,205)( 85,215)( 86,225)( 87,235)( 88,124)( 89,145)( 90,155)( 91,176)( 92,186)( 93,196)( 94,206)( 95,216)( 96,226)( 97,236)( 98,125)( 99,135)(100,156)(101,166)(102,187)(103,197)(104,207)(105,217)(106,227)(107,237)(108,126)(109,136)(110,146)(111,167)(112,177)(113,198)(114,208)(115,218)(116,228)(117,238)(118,127)(119,137)(120,147)(121,157);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(242)!(  2,112)(  3,102)(  4, 92)(  5, 82)(  6, 72)(  7, 62)(  8, 52)(  9, 42)( 10, 32)( 11, 22)( 12,111)( 13,101)( 14, 91)( 15, 81)( 16, 71)( 17, 61)( 18, 51)( 19, 41)( 20, 31)( 23,100)( 24, 90)( 25, 80)( 26, 70)( 27, 60)( 28, 50)( 29, 40)( 33,121)( 34, 89)( 35, 79)( 36, 69)( 37, 59)( 38, 49)( 43,120)( 44,110)( 45, 78)( 46, 68)( 47, 58)( 53,119)( 54,109)( 55, 99)( 56, 67)( 63,118)( 64,108)( 65, 98)( 66, 88)( 73,117)( 74,107)( 75, 97)( 76, 87)( 83,116)( 84,106)( 85, 96)( 93,115)( 94,105)(103,114)(123,233)(124,223)(125,213)(126,203)(127,193)(128,183)(129,173)(130,163)(131,153)(132,143)(133,232)(134,222)(135,212)(136,202)(137,192)(138,182)(139,172)(140,162)(141,152)(144,221)(145,211)(146,201)(147,191)(148,181)(149,171)(150,161)(154,242)(155,210)(156,200)(157,190)(158,180)(159,170)(164,241)(165,231)(166,199)(167,189)(168,179)(174,240)(175,230)(176,220)(177,188)(184,239)(185,229)(186,219)(187,209)(194,238)(195,228)(196,218)(197,208)(204,237)(205,227)(206,217)(214,236)(215,226)(224,235);
s1 := Sym(242)!( 12,113)( 13,114)( 14,115)( 15,116)( 16,117)( 17,118)( 18,119)( 19,120)( 20,121)( 21,111)( 22,112)( 23,104)( 24,105)( 25,106)( 26,107)( 27,108)( 28,109)( 29,110)( 30,100)( 31,101)( 32,102)( 33,103)( 34, 95)( 35, 96)( 36, 97)( 37, 98)( 38, 99)( 39, 89)( 40, 90)( 41, 91)( 42, 92)( 43, 93)( 44, 94)( 45, 86)( 46, 87)( 47, 88)( 48, 78)( 49, 79)( 50, 80)( 51, 81)( 52, 82)( 53, 83)( 54, 84)( 55, 85)( 56, 77)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 61, 71)( 62, 72)( 63, 73)( 64, 74)( 65, 75)( 66, 76)(133,234)(134,235)(135,236)(136,237)(137,238)(138,239)(139,240)(140,241)(141,242)(142,232)(143,233)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,221)(152,222)(153,223)(154,224)(155,216)(156,217)(157,218)(158,219)(159,220)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,207)(167,208)(168,209)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(175,205)(176,206)(177,198)(178,188)(179,189)(180,190)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197);
s2 := Sym(242)!(  1,178)(  2,188)(  3,209)(  4,219)(  5,229)(  6,239)(  7,128)(  8,138)(  9,148)( 10,158)( 11,168)( 12,189)( 13,199)( 14,220)( 15,230)( 16,240)( 17,129)( 18,139)( 19,149)( 20,159)( 21,169)( 22,179)( 23,200)( 24,210)( 25,231)( 26,241)( 27,130)( 28,140)( 29,150)( 30,160)( 31,170)( 32,180)( 33,190)( 34,211)( 35,221)( 36,242)( 37,131)( 38,141)( 39,151)( 40,161)( 41,171)( 42,181)( 43,191)( 44,201)( 45,222)( 46,232)( 47,132)( 48,142)( 49,152)( 50,162)( 51,172)( 52,182)( 53,192)( 54,202)( 55,212)( 56,233)( 57,122)( 58,143)( 59,153)( 60,163)( 61,173)( 62,183)( 63,193)( 64,203)( 65,213)( 66,223)( 67,123)( 68,133)( 69,154)( 70,164)( 71,174)( 72,184)( 73,194)( 74,204)( 75,214)( 76,224)( 77,234)( 78,134)( 79,144)( 80,165)( 81,175)( 82,185)( 83,195)( 84,205)( 85,215)( 86,225)( 87,235)( 88,124)( 89,145)( 90,155)( 91,176)( 92,186)( 93,196)( 94,206)( 95,216)( 96,226)( 97,236)( 98,125)( 99,135)(100,156)(101,166)(102,187)(103,197)(104,207)(105,217)(106,227)(107,237)(108,126)(109,136)(110,146)(111,167)(112,177)(113,198)(114,208)(115,218)(116,228)(117,238)(118,127)(119,137)(120,147)(121,157);
poly := sub<Sym(242)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle