Polytope of Type {4,28,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,28,4}*1792b
if this polytope has a name.
Group : SmallGroup(1792,201150)
Rank : 4
Schlafli Type : {4,28,4}
Number of vertices, edges, etc : 8, 112, 112, 4
Order of s₀s₁s₂s₃ : 28
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, {4,28,2}*896
   4-fold quotients : {2,28,4}*448, {4,28,2}*448, {4,14,4}*448
   7-fold quotients : {4,4,4}*256b
   8-fold quotients : {2,28,2}*224, {2,14,4}*224, {4,14,2}*224
   14-fold quotients : {4,4,4}*128, {4,4,2}*128
   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, 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);;
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)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(113,141)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,148)
(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,155)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,162)(135,168)(136,167)
(137,166)(138,165)(139,164)(140,163)(169,197)(170,203)(171,202)(172,201)
(173,200)(174,199)(175,198)(176,204)(177,210)(178,209)(179,208)(180,207)
(181,206)(182,205)(183,211)(184,217)(185,216)(186,215)(187,214)(188,213)
(189,212)(190,218)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219);;
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)( 29,142)( 30,141)( 31,147)( 32,146)
( 33,145)( 34,144)( 35,143)( 36,149)( 37,148)( 38,154)( 39,153)( 40,152)
( 41,151)( 42,150)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)( 56,164)
( 57,170)( 58,169)( 59,175)( 60,174)( 61,173)( 62,172)( 63,171)( 64,177)
( 65,176)( 66,182)( 67,181)( 68,180)( 69,179)( 70,178)( 71,184)( 72,183)
( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,191)( 79,190)( 80,196)
( 81,195)( 82,194)( 83,193)( 84,192)( 85,198)( 86,197)( 87,203)( 88,202)
( 89,201)( 90,200)( 91,199)( 92,205)( 93,204)( 94,210)( 95,209)( 96,208)
( 97,207)( 98,206)( 99,212)(100,211)(101,217)(102,216)(103,215)(104,214)
(105,213)(106,219)(107,218)(108,224)(109,223)(110,222)(111,221)(112,220);;
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, 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);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
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*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, 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);
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)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(113,141)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)
(120,148)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,155)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,162)(135,168)
(136,167)(137,166)(138,165)(139,164)(140,163)(169,197)(170,203)(171,202)
(172,201)(173,200)(174,199)(175,198)(176,204)(177,210)(178,209)(179,208)
(180,207)(181,206)(182,205)(183,211)(184,217)(185,216)(186,215)(187,214)
(188,213)(189,212)(190,218)(191,224)(192,223)(193,222)(194,221)(195,220)
(196,219);
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)( 29,142)( 30,141)( 31,147)
( 32,146)( 33,145)( 34,144)( 35,143)( 36,149)( 37,148)( 38,154)( 39,153)
( 40,152)( 41,151)( 42,150)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)
( 56,164)( 57,170)( 58,169)( 59,175)( 60,174)( 61,173)( 62,172)( 63,171)
( 64,177)( 65,176)( 66,182)( 67,181)( 68,180)( 69,179)( 70,178)( 71,184)
( 72,183)( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,191)( 79,190)
( 80,196)( 81,195)( 82,194)( 83,193)( 84,192)( 85,198)( 86,197)( 87,203)
( 88,202)( 89,201)( 90,200)( 91,199)( 92,205)( 93,204)( 94,210)( 95,209)
( 96,208)( 97,207)( 98,206)( 99,212)(100,211)(101,217)(102,216)(103,215)
(104,214)(105,213)(106,219)(107,218)(108,224)(109,223)(110,222)(111,221)
(112,220);
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, 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);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
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*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope