Part of the Atlas of Small Regular Polytopes

Polytope of Type {28,4}

Atlas Canonical Name {28,4}*1568

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1568,821)
Rank
3
Schläfli Type
{28,4}
Vertices, edges, …
196, 392, 28
Order of s0s1s2
4
Order of s0s1s2s1
14
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

49-fold

98-fold

196-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)^7*s2*(s1*s0)^6*s1*s2> of order 2

14 facets

98 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2> of order 7

4 facets

28 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^5*s1*s2*s1> of order 7

4 facets

28 vertex figures

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

16 facets

28 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle