Polytope of Type {4,56,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,56,4}*1792b
if this polytope has a name.
Group : SmallGroup(1792,201204)
Rank : 4
Schlafli Type : {4,56,4}
Number of vertices, edges, etc : 4, 112, 112, 4
Order of s₀s₁s₂s₃ : 56
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
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 : {4,28,4}*896, {2,56,4}*896b
   4-fold quotients : {2,28,4}*448, {4,28,2}*448, {4,14,4}*448
   7-fold quotients : {4,8,4}*256b
   8-fold quotients : {2,28,2}*224, {2,14,4}*224, {4,14,2}*224
   14-fold quotients : {4,4,4}*128, {2,8,4}*128b
   16-fold quotients : {2,14,2}*112
   28-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   32-fold quotients : {2,7,2}*56
   56-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
   112-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)(  8, 64)
(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)
( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)( 24, 80)
( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 85)( 30, 86)( 31, 87)( 32, 88)
( 33, 89)( 34, 90)( 35, 91)( 36, 92)( 37, 93)( 38, 94)( 39, 95)( 40, 96)
( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,101)( 46,102)( 47,103)( 48,104)
( 49,105)( 50,106)( 51,107)( 52,108)( 53,109)( 54,110)( 55,111)( 56,112)
(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)
(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,169)(128,170)
(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)
(137,179)(138,180)(139,181)(140,182)(141,211)(142,212)(143,213)(144,214)
(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)
(153,223)(154,224)(155,197)(156,198)(157,199)(158,200)(159,201)(160,202)
(161,203)(162,204)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210);;
s1 := (  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)
( 57,169)( 58,175)( 59,174)( 60,173)( 61,172)( 62,171)( 63,170)( 64,176)
( 65,182)( 66,181)( 67,180)( 68,179)( 69,178)( 70,177)( 71,183)( 72,189)
( 73,188)( 74,187)( 75,186)( 76,185)( 77,184)( 78,190)( 79,196)( 80,195)
( 81,194)( 82,193)( 83,192)( 84,191)( 85,204)( 86,210)( 87,209)( 88,208)
( 89,207)( 90,206)( 91,205)( 92,197)( 93,203)( 94,202)( 95,201)( 96,200)
( 97,199)( 98,198)( 99,218)(100,224)(101,223)(102,222)(103,221)(104,220)
(105,219)(106,211)(107,217)(108,216)(109,215)(110,214)(111,213)(112,212);;
s2 := (  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, 37)( 30, 36)( 31, 42)( 32, 41)
( 33, 40)( 34, 39)( 35, 38)( 43, 51)( 44, 50)( 45, 56)( 46, 55)( 47, 54)
( 48, 53)( 49, 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, 93)( 86, 92)
( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,107)(100,106)(101,112)
(102,111)(103,110)(104,109)(105,108)(113,142)(114,141)(115,147)(116,146)
(117,145)(118,144)(119,143)(120,149)(121,148)(122,154)(123,153)(124,152)
(125,151)(126,150)(127,156)(128,155)(129,161)(130,160)(131,159)(132,158)
(133,157)(134,163)(135,162)(136,168)(137,167)(138,166)(139,165)(140,164)
(169,198)(170,197)(171,203)(172,202)(173,201)(174,200)(175,199)(176,205)
(177,204)(178,210)(179,209)(180,208)(181,207)(182,206)(183,212)(184,211)
(185,217)(186,216)(187,215)(188,214)(189,213)(190,219)(191,218)(192,224)
(193,223)(194,222)(195,221)(196,220);;
s3 := (  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)(  8, 64)
(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)
( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)( 24, 80)
( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 92)( 30, 93)( 31, 94)( 32, 95)
( 33, 96)( 34, 97)( 35, 98)( 36, 85)( 37, 86)( 38, 87)( 39, 88)( 40, 89)
( 41, 90)( 42, 91)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)( 48,111)
( 49,112)( 50, 99)( 51,100)( 52,101)( 53,102)( 54,103)( 55,104)( 56,105)
(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,176)
(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,183)(128,184)
(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)
(137,193)(138,194)(139,195)(140,196)(141,204)(142,205)(143,206)(144,207)
(145,208)(146,209)(147,210)(148,197)(149,198)(150,199)(151,200)(152,201)
(153,202)(154,203)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)
(161,224)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(224)!(  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)
(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)
( 16, 72)( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)
( 24, 80)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 85)( 30, 86)( 31, 87)
( 32, 88)( 33, 89)( 34, 90)( 35, 91)( 36, 92)( 37, 93)( 38, 94)( 39, 95)
( 40, 96)( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,101)( 46,102)( 47,103)
( 48,104)( 49,105)( 50,106)( 51,107)( 52,108)( 53,109)( 54,110)( 55,111)
( 56,112)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)
(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,169)
(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)
(136,178)(137,179)(138,180)(139,181)(140,182)(141,211)(142,212)(143,213)
(144,214)(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)
(152,222)(153,223)(154,224)(155,197)(156,198)(157,199)(158,200)(159,201)
(160,202)(161,203)(162,204)(163,205)(164,206)(165,207)(166,208)(167,209)
(168,210);
s1 := 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)( 57,169)( 58,175)( 59,174)( 60,173)( 61,172)( 62,171)( 63,170)
( 64,176)( 65,182)( 66,181)( 67,180)( 68,179)( 69,178)( 70,177)( 71,183)
( 72,189)( 73,188)( 74,187)( 75,186)( 76,185)( 77,184)( 78,190)( 79,196)
( 80,195)( 81,194)( 82,193)( 83,192)( 84,191)( 85,204)( 86,210)( 87,209)
( 88,208)( 89,207)( 90,206)( 91,205)( 92,197)( 93,203)( 94,202)( 95,201)
( 96,200)( 97,199)( 98,198)( 99,218)(100,224)(101,223)(102,222)(103,221)
(104,220)(105,219)(106,211)(107,217)(108,216)(109,215)(110,214)(111,213)
(112,212);
s2 := 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, 37)( 30, 36)( 31, 42)
( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 51)( 44, 50)( 45, 56)( 46, 55)
( 47, 54)( 48, 53)( 49, 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, 93)
( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,107)(100,106)
(101,112)(102,111)(103,110)(104,109)(105,108)(113,142)(114,141)(115,147)
(116,146)(117,145)(118,144)(119,143)(120,149)(121,148)(122,154)(123,153)
(124,152)(125,151)(126,150)(127,156)(128,155)(129,161)(130,160)(131,159)
(132,158)(133,157)(134,163)(135,162)(136,168)(137,167)(138,166)(139,165)
(140,164)(169,198)(170,197)(171,203)(172,202)(173,201)(174,200)(175,199)
(176,205)(177,204)(178,210)(179,209)(180,208)(181,207)(182,206)(183,212)
(184,211)(185,217)(186,216)(187,215)(188,214)(189,213)(190,219)(191,218)
(192,224)(193,223)(194,222)(195,221)(196,220);
s3 := Sym(224)!(  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)
(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)
( 16, 72)( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)
( 24, 80)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 92)( 30, 93)( 31, 94)
( 32, 95)( 33, 96)( 34, 97)( 35, 98)( 36, 85)( 37, 86)( 38, 87)( 39, 88)
( 40, 89)( 41, 90)( 42, 91)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)
( 48,111)( 49,112)( 50, 99)( 51,100)( 52,101)( 53,102)( 54,103)( 55,104)
( 56,105)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)
(120,176)(121,177)(122,178)(123,179)(124,180)(125,181)(126,182)(127,183)
(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)
(136,192)(137,193)(138,194)(139,195)(140,196)(141,204)(142,205)(143,206)
(144,207)(145,208)(146,209)(147,210)(148,197)(149,198)(150,199)(151,200)
(152,201)(153,202)(154,203)(155,218)(156,219)(157,220)(158,221)(159,222)
(160,223)(161,224)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)
(168,217);
poly := sub<Sym(224)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s3*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*s3*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 >;

References : None.
to this polytope