Polytope of Type {56,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,4}*1792b
if this polytope has a name.
Group : SmallGroup(1792,90280)
Rank : 3
Schlafli Type : {56,4}
Number of vertices, edges, etc : 224, 448, 16
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {56,4}*896b
   4-fold quotients : {28,4}*448
   7-fold quotients : {8,4}*256b
   8-fold quotients : {28,4}*224
   14-fold quotients : {8,4}*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.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope