Polytope of Type {4,56}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,56}*1792b
if this polytope has a name.
Group : SmallGroup(1792,90280)
Rank : 3
Schlafli Type : {4,56}
Number of vertices, edges, etc : 16, 448, 224
Order of s0s1s2 : 28
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,56}*896b
   4-fold quotients : {4,28}*448
   7-fold quotients : {4,8}*256b
   8-fold quotients : {4,28}*224
   14-fold quotients : {4,8}*128b
   16-fold quotients : {2,28}*112, {4,14}*112
   28-fold quotients : {4,4}*64
   32-fold quotients : {2,14}*56
   56-fold quotients : {4,4}*32
   64-fold quotients : {2,7}*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=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      112 facets:
         112 of {4}*8
      12 vertex figures:
         8 of {28}*56
         4 of {56}*112
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      56 facets:
         56 of {4}*8
      6 vertex figures:
         4 of {28}*56
         2 of {56}*112
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      56 facets:
         56 of {4}*8
      10 vertex figures:
         8 of {14}*28
         2 of {56}*112

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