Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,4}

Atlas Canonical Name {10,4}*2000a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(2000,482)
Rank
3
Schläfli Type
{10,4}
Vertices, edges, …
250, 500, 100
Order of s0s1s2
20
Order of s0s1s2s1
10
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

5-fold

10-fold

125-fold

250-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*s0)^4*s1*s2*s1> of order 2

51 facets

125 vertex figures

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

50 facets

130 vertex figures

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

26 facets

65 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 5

28 facets

50 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2> of order 5

20 facets

50 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 5

20 facets

50 vertex figures

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

14 facets

30 vertex figures

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

10 facets

30 vertex figures

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

15 facets

25 vertex figures

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

10 facets

30 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1,126)(  2,127)(  3,128)(  4,129)(  5,130)(  6,146)(  7,147)(  8,148)(  9,149)( 10,150)( 11,141)( 12,142)( 13,143)( 14,144)( 15,145)( 16,136)( 17,137)( 18,138)( 19,139)( 20,140)( 21,131)( 22,132)( 23,133)( 24,134)( 25,135)( 26,229)( 27,230)( 28,226)( 29,227)( 30,228)( 31,249)( 32,250)( 33,246)( 34,247)( 35,248)( 36,244)( 37,245)( 38,241)( 39,242)( 40,243)( 41,239)( 42,240)( 43,236)( 44,237)( 45,238)( 46,234)( 47,235)( 48,231)( 49,232)( 50,233)( 51,202)( 52,203)( 53,204)( 54,205)( 55,201)( 56,222)( 57,223)( 58,224)( 59,225)( 60,221)( 61,217)( 62,218)( 63,219)( 64,220)( 65,216)( 66,212)( 67,213)( 68,214)( 69,215)( 70,211)( 71,207)( 72,208)( 73,209)( 74,210)( 75,206)( 76,180)( 77,176)( 78,177)( 79,178)( 80,179)( 81,200)( 82,196)( 83,197)( 84,198)( 85,199)( 86,195)( 87,191)( 88,192)( 89,193)( 90,194)( 91,190)( 92,186)( 93,187)( 94,188)( 95,189)( 96,185)( 97,181)( 98,182)( 99,183)(100,184)(101,153)(102,154)(103,155)(104,151)(105,152)(106,173)(107,174)(108,175)(109,171)(110,172)(111,168)(112,169)(113,170)(114,166)(115,167)(116,163)(117,164)(118,165)(119,161)(120,162)(121,158)(122,159)(123,160)(124,156)(125,157);;
s1 := (  1, 26)(  2, 30)(  3, 29)(  4, 28)(  5, 27)(  6, 85)(  7, 84)(  8, 83)(  9, 82)( 10, 81)( 12, 15)( 13, 14)( 16, 69)( 17, 68)( 18, 67)( 19, 66)( 20, 70)( 21,124)( 22,123)( 23,122)( 24,121)( 25,125)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36,111)( 37,115)( 38,114)( 39,113)( 40,112)( 41, 44)( 42, 43)( 46, 99)( 47, 98)( 48, 97)( 49, 96)( 50,100)( 51,101)( 52,105)( 53,104)( 54,103)( 55,102)( 61, 86)( 62, 90)( 63, 89)( 64, 88)( 65, 87)( 71, 74)( 72, 73)( 77, 80)( 78, 79)( 91,119)( 92,118)( 93,117)( 94,116)( 95,120)(106,110)(107,109)(126,151)(127,155)(128,154)(129,153)(130,152)(131,210)(132,209)(133,208)(134,207)(135,206)(137,140)(138,139)(141,194)(142,193)(143,192)(144,191)(145,195)(146,249)(147,248)(148,247)(149,246)(150,250)(156,185)(157,184)(158,183)(159,182)(160,181)(161,236)(162,240)(163,239)(164,238)(165,237)(166,169)(167,168)(171,224)(172,223)(173,222)(174,221)(175,225)(176,226)(177,230)(178,229)(179,228)(180,227)(186,211)(187,215)(188,214)(189,213)(190,212)(196,199)(197,198)(202,205)(203,204)(216,244)(217,243)(218,242)(219,241)(220,245)(231,235)(232,234);;
s2 := (  1,  2)(  3,  5)(  6,115)(  7,114)(  8,113)(  9,112)( 10,111)( 11, 96)( 12,100)( 13, 99)( 14, 98)( 15, 97)( 16, 60)( 17, 59)( 18, 58)( 19, 57)( 20, 56)( 21, 42)( 22, 41)( 23, 45)( 24, 44)( 25, 43)( 26, 93)( 27, 92)( 28, 91)( 29, 95)( 30, 94)( 31, 53)( 32, 52)( 33, 51)( 34, 55)( 35, 54)( 37, 40)( 38, 39)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 61,105)( 62,104)( 63,103)( 64,102)( 65,101)( 66, 88)( 67, 87)( 68, 86)( 69, 90)( 70, 89)( 71, 74)( 72, 73)( 76,122)( 77,121)( 78,125)( 79,124)( 80,123)( 82, 85)( 83, 84)(116,117)(118,120)(126,127)(128,130)(131,240)(132,239)(133,238)(134,237)(135,236)(136,221)(137,225)(138,224)(139,223)(140,222)(141,185)(142,184)(143,183)(144,182)(145,181)(146,167)(147,166)(148,170)(149,169)(150,168)(151,218)(152,217)(153,216)(154,220)(155,219)(156,178)(157,177)(158,176)(159,180)(160,179)(162,165)(163,164)(171,231)(172,235)(173,234)(174,233)(175,232)(186,230)(187,229)(188,228)(189,227)(190,226)(191,213)(192,212)(193,211)(194,215)(195,214)(196,199)(197,198)(201,247)(202,246)(203,250)(204,249)(205,248)(207,210)(208,209)(241,242)(243,245);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(250)!(  1,126)(  2,127)(  3,128)(  4,129)(  5,130)(  6,146)(  7,147)(  8,148)(  9,149)( 10,150)( 11,141)( 12,142)( 13,143)( 14,144)( 15,145)( 16,136)( 17,137)( 18,138)( 19,139)( 20,140)( 21,131)( 22,132)( 23,133)( 24,134)( 25,135)( 26,229)( 27,230)( 28,226)( 29,227)( 30,228)( 31,249)( 32,250)( 33,246)( 34,247)( 35,248)( 36,244)( 37,245)( 38,241)( 39,242)( 40,243)( 41,239)( 42,240)( 43,236)( 44,237)( 45,238)( 46,234)( 47,235)( 48,231)( 49,232)( 50,233)( 51,202)( 52,203)( 53,204)( 54,205)( 55,201)( 56,222)( 57,223)( 58,224)( 59,225)( 60,221)( 61,217)( 62,218)( 63,219)( 64,220)( 65,216)( 66,212)( 67,213)( 68,214)( 69,215)( 70,211)( 71,207)( 72,208)( 73,209)( 74,210)( 75,206)( 76,180)( 77,176)( 78,177)( 79,178)( 80,179)( 81,200)( 82,196)( 83,197)( 84,198)( 85,199)( 86,195)( 87,191)( 88,192)( 89,193)( 90,194)( 91,190)( 92,186)( 93,187)( 94,188)( 95,189)( 96,185)( 97,181)( 98,182)( 99,183)(100,184)(101,153)(102,154)(103,155)(104,151)(105,152)(106,173)(107,174)(108,175)(109,171)(110,172)(111,168)(112,169)(113,170)(114,166)(115,167)(116,163)(117,164)(118,165)(119,161)(120,162)(121,158)(122,159)(123,160)(124,156)(125,157);
s1 := Sym(250)!(  1, 26)(  2, 30)(  3, 29)(  4, 28)(  5, 27)(  6, 85)(  7, 84)(  8, 83)(  9, 82)( 10, 81)( 12, 15)( 13, 14)( 16, 69)( 17, 68)( 18, 67)( 19, 66)( 20, 70)( 21,124)( 22,123)( 23,122)( 24,121)( 25,125)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36,111)( 37,115)( 38,114)( 39,113)( 40,112)( 41, 44)( 42, 43)( 46, 99)( 47, 98)( 48, 97)( 49, 96)( 50,100)( 51,101)( 52,105)( 53,104)( 54,103)( 55,102)( 61, 86)( 62, 90)( 63, 89)( 64, 88)( 65, 87)( 71, 74)( 72, 73)( 77, 80)( 78, 79)( 91,119)( 92,118)( 93,117)( 94,116)( 95,120)(106,110)(107,109)(126,151)(127,155)(128,154)(129,153)(130,152)(131,210)(132,209)(133,208)(134,207)(135,206)(137,140)(138,139)(141,194)(142,193)(143,192)(144,191)(145,195)(146,249)(147,248)(148,247)(149,246)(150,250)(156,185)(157,184)(158,183)(159,182)(160,181)(161,236)(162,240)(163,239)(164,238)(165,237)(166,169)(167,168)(171,224)(172,223)(173,222)(174,221)(175,225)(176,226)(177,230)(178,229)(179,228)(180,227)(186,211)(187,215)(188,214)(189,213)(190,212)(196,199)(197,198)(202,205)(203,204)(216,244)(217,243)(218,242)(219,241)(220,245)(231,235)(232,234);
s2 := Sym(250)!(  1,  2)(  3,  5)(  6,115)(  7,114)(  8,113)(  9,112)( 10,111)( 11, 96)( 12,100)( 13, 99)( 14, 98)( 15, 97)( 16, 60)( 17, 59)( 18, 58)( 19, 57)( 20, 56)( 21, 42)( 22, 41)( 23, 45)( 24, 44)( 25, 43)( 26, 93)( 27, 92)( 28, 91)( 29, 95)( 30, 94)( 31, 53)( 32, 52)( 33, 51)( 34, 55)( 35, 54)( 37, 40)( 38, 39)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 61,105)( 62,104)( 63,103)( 64,102)( 65,101)( 66, 88)( 67, 87)( 68, 86)( 69, 90)( 70, 89)( 71, 74)( 72, 73)( 76,122)( 77,121)( 78,125)( 79,124)( 80,123)( 82, 85)( 83, 84)(116,117)(118,120)(126,127)(128,130)(131,240)(132,239)(133,238)(134,237)(135,236)(136,221)(137,225)(138,224)(139,223)(140,222)(141,185)(142,184)(143,183)(144,182)(145,181)(146,167)(147,166)(148,170)(149,169)(150,168)(151,218)(152,217)(153,216)(154,220)(155,219)(156,178)(157,177)(158,176)(159,180)(160,179)(162,165)(163,164)(171,231)(172,235)(173,234)(174,233)(175,232)(186,230)(187,229)(188,228)(189,227)(190,226)(191,213)(192,212)(193,211)(194,215)(195,214)(196,199)(197,198)(201,247)(202,246)(203,250)(204,249)(205,248)(207,210)(208,209)(241,242)(243,245);
poly := sub<Sym(250)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle