Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*1600

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1600,6624)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
200, 400, 200
Order of s0s1s2
20
Order of s0s1s2s1
20
Also known as
{4,4}(10,10), {4,4}20. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

25-fold

50-fold

100-fold

200-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*s2*s1)^10> of order 2

100 facets

100 vertex figures

P/N, where N=<(s0*s1*s2*s1)^9*s0*s1*s2> of order 2

100 facets

100 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^9*s2> of order 2

100 facets

102 vertex figures

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

102 facets

100 vertex figures

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^9*s0*s1*s2> of order 4

51 facets

50 vertex figures

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

50 facets

50 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^9*s2> of order 4

51 facets

51 vertex figures

P/N, where N=<(s0*s2*s1)^10, (s0*s1)^2*(s2*s1*s0)^8*s2*s1*s2> of order 4

50 facets

50 vertex figures

P/N, where N=<s1*(s2*s1*s0)^8*s2*s1*s2, s0*s1*(s2*s1*s0)^9*s2*s1> of order 4

50 facets

51 vertex figures

P/N, where N=<(s0*s1*s2*s1)^4> of order 5

40 facets

40 vertex figures

P/N, where N=<(s0*s2*s1)^4> of order 5

40 facets

40 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1*s0*s1)^4*(s2*s1*s0)^2*s2*s1*s2> of order 5

40 facets

40 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^3*s2> of order 10

22 facets

20 vertex figures

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

20 facets

20 vertex figures

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

22 facets

20 vertex figures

P/N, where N=<(s0*s1)^2, s0*(s2*s1*s0*s1)^3*(s2*s1*s0)^4*s2> of order 10

22 facets

20 vertex figures

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

20 facets

20 vertex figures

P/N, where N=<(s0*s2*s1)^4, s0*(s1*s0*s2)^9*s1*s2> of order 10

20 facets

20 vertex figures

P/N, where N=<(s1*s2*s1*s0)^2*(s1*s2)^2, s0*s1*s0*s2*s1*s0*s1*(s2*s1*s0)^3> of order 10

20 facets

22 vertex figures

P/N, where N=<(s0*s1*s2*s1)^4, s0*(s2*s1*s0*s1)^3*(s2*s1*s0)^2*s2> of order 10

20 facets

22 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 10

20 facets

22 vertex figures

P/N, where N=<(s0*s2*s1)^4, s0*s1*s0*s2*s1*s0*s1*(s2*s1*s0)^7*s2*s1*s2> of order 10

20 facets

20 vertex figures

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^4, s0*(s2*s1*s0*s1)^3*(s2*s1*s0)^2*s2> of order 20

11 facets

11 vertex figures

P/N, where N=<(s0*s2*s1)^4, (s0*s1)^2*s2*s1*s0*s1*(s2*s1*s0)^2*s2*s1*s2> of order 20

10 facets

10 vertex figures

P/N, where N=<(s0*s1)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 20

11 facets

11 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^3*s2, s1*(s2*s1*s0)^8*s2*s1*s2> of order 20

11 facets

11 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle