Polytope of Type {28,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,8}*1792b
if this polytope has a name.
Group : SmallGroup(1792,90281)
Rank : 3
Schlafli Type : {28,8}
Number of vertices, edges, etc : 112, 448, 32
Order of s0s1s2 : 28
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {28,8}*896b
   4-fold quotients : {28,4}*448
   7-fold quotients : {4,8}*256b
   8-fold quotients : {28,4}*224
   14-fold quotients : {4,8}*128b
   16-fold quotients : {28,2}*112, {14,4}*112
   28-fold quotients : {4,4}*64
   32-fold quotients : {14,2}*56
   56-fold quotients : {4,4}*32
   64-fold quotients : {7,2}*28
   112-fold quotients : {2,4}*16, {4,2}*16
   224-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      16 facets:
         16 of {28}*56
      84 vertex figures:
         56 of {4}*8
         28 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
      8 facets:
         8 of {28}*56
      42 vertex figures:
         28 of {4}*8
         14 of {8}*16
   P/N, where N=<s1*s2*s1*s2> of order 4.
      8 facets:
         8 of {28}*56
      70 vertex figures:
         56 of {2}*4
         14 of {8}*16

Permutation Representation (GAP) : 
s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 30, 35)( 31, 34)( 32, 33)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)( 46, 54)( 47, 53)( 48, 52)( 49, 51)( 57,106)( 58,112)( 59,111)( 60,110)( 61,109)( 62,108)( 63,107)( 64, 99)( 65,105)( 66,104)( 67,103)( 68,102)( 69,101)( 70,100)( 71, 85)( 72, 91)( 73, 90)( 74, 89)( 75, 88)( 76, 87)( 77, 86)( 78, 92)( 79, 98)( 80, 97)( 81, 96)( 82, 95)( 83, 94)( 84, 93)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(127,134)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(142,147)(143,146)(144,145)(149,154)(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(169,218)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,211)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,197)(184,203)(185,202)(186,201)(187,200)(188,199)(189,198)(190,204)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205);;
s1 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 51)( 37, 50)( 38, 56)( 39, 55)( 40, 54)( 41, 53)( 42, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)(137,195)(138,194)(139,193)(140,192)(141,212)(142,211)(143,217)(144,216)(145,215)(146,214)(147,213)(148,219)(149,218)(150,224)(151,223)(152,222)(153,221)(154,220)(155,198)(156,197)(157,203)(158,202)(159,201)(160,200)(161,199)(162,205)(163,204)(164,210)(165,209)(166,208)(167,207)(168,206);;
s2 := (  1,113)(  2,114)(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,120)(  9,121)( 10,122)( 11,123)( 12,124)( 13,125)( 14,126)( 15,127)( 16,128)( 17,129)( 18,130)( 19,131)( 20,132)( 21,133)( 22,134)( 23,135)( 24,136)( 25,137)( 26,138)( 27,139)( 28,140)( 29,148)( 30,149)( 31,150)( 32,151)( 33,152)( 34,153)( 35,154)( 36,141)( 37,142)( 38,143)( 39,144)( 40,145)( 41,146)( 42,147)( 43,162)( 44,163)( 45,164)( 46,165)( 47,166)( 48,167)( 49,168)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)( 57,218)( 58,219)( 59,220)( 60,221)( 61,222)( 62,223)( 63,224)( 64,211)( 65,212)( 66,213)( 67,214)( 68,215)( 69,216)( 70,217)( 71,204)( 72,205)( 73,206)( 74,207)( 75,208)( 76,209)( 77,210)( 78,197)( 79,198)( 80,199)( 81,200)( 82,201)( 83,202)( 84,203)( 85,190)( 86,191)( 87,192)( 88,193)( 89,194)( 90,195)( 91,196)( 92,183)( 93,184)( 94,185)( 95,186)( 96,187)( 97,188)( 98,189)( 99,176)(100,177)(101,178)(102,179)(103,180)(104,181)(105,182)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175);;
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*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(224)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 30, 35)( 31, 34)( 32, 33)( 37, 42)( 38, 41)( 39, 40)( 43, 50)( 44, 56)( 45, 55)( 46, 54)( 47, 53)( 48, 52)( 49, 51)( 57,106)( 58,112)( 59,111)( 60,110)( 61,109)( 62,108)( 63,107)( 64, 99)( 65,105)( 66,104)( 67,103)( 68,102)( 69,101)( 70,100)( 71, 85)( 72, 91)( 73, 90)( 74, 89)( 75, 88)( 76, 87)( 77, 86)( 78, 92)( 79, 98)( 80, 97)( 81, 96)( 82, 95)( 83, 94)( 84, 93)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(127,134)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(142,147)(143,146)(144,145)(149,154)(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(169,218)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,211)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,197)(184,203)(185,202)(186,201)(187,200)(188,199)(189,198)(190,204)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205);
s1 := Sym(224)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 51)( 37, 50)( 38, 56)( 39, 55)( 40, 54)( 41, 53)( 42, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)(137,195)(138,194)(139,193)(140,192)(141,212)(142,211)(143,217)(144,216)(145,215)(146,214)(147,213)(148,219)(149,218)(150,224)(151,223)(152,222)(153,221)(154,220)(155,198)(156,197)(157,203)(158,202)(159,201)(160,200)(161,199)(162,205)(163,204)(164,210)(165,209)(166,208)(167,207)(168,206);
s2 := Sym(224)!(  1,113)(  2,114)(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,120)(  9,121)( 10,122)( 11,123)( 12,124)( 13,125)( 14,126)( 15,127)( 16,128)( 17,129)( 18,130)( 19,131)( 20,132)( 21,133)( 22,134)( 23,135)( 24,136)( 25,137)( 26,138)( 27,139)( 28,140)( 29,148)( 30,149)( 31,150)( 32,151)( 33,152)( 34,153)( 35,154)( 36,141)( 37,142)( 38,143)( 39,144)( 40,145)( 41,146)( 42,147)( 43,162)( 44,163)( 45,164)( 46,165)( 47,166)( 48,167)( 49,168)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)( 57,218)( 58,219)( 59,220)( 60,221)( 61,222)( 62,223)( 63,224)( 64,211)( 65,212)( 66,213)( 67,214)( 68,215)( 69,216)( 70,217)( 71,204)( 72,205)( 73,206)( 74,207)( 75,208)( 76,209)( 77,210)( 78,197)( 79,198)( 80,199)( 81,200)( 82,201)( 83,202)( 84,203)( 85,190)( 86,191)( 87,192)( 88,193)( 89,194)( 90,195)( 91,196)( 92,183)( 93,184)( 94,185)( 95,186)( 96,187)( 97,188)( 98,189)( 99,176)(100,177)(101,178)(102,179)(103,180)(104,181)(105,182)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175);
poly := sub<Sym(224)|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*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*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*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 >;
References : None.
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