Polytope of Type {28,8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,8,4}*1792c
if this polytope has a name.
Group : SmallGroup(1792,201205)
Rank : 4
Schlafli Type : {28,8,4}
Number of vertices, edges, etc : 28, 112, 16, 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 : {28,4,4}*896, {28,8,2}*896b
   4-fold quotients : {28,4,2}*448, {28,2,4}*448, {14,4,4}*448
   7-fold quotients : {4,8,4}*256c
   8-fold quotients : {28,2,2}*224, {14,2,4}*224, {14,4,2}*224
   14-fold quotients : {4,4,4}*128, {4,8,2}*128b
   16-fold quotients : {7,2,4}*112, {14,2,2}*112
   28-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   32-fold quotients : {7,2,2}*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, 63)(  3, 62)(  4, 61)(  5, 60)(  6, 59)(  7, 58)(  8, 64)
(  9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 71)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 78)( 23, 84)( 24, 83)
( 25, 82)( 26, 81)( 27, 80)( 28, 79)( 29, 92)( 30, 98)( 31, 97)( 32, 96)
( 33, 95)( 34, 94)( 35, 93)( 36, 85)( 37, 91)( 38, 90)( 39, 89)( 40, 88)
( 41, 87)( 42, 86)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)( 48,108)
( 49,107)( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)( 56,100)
(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)(141,204)(142,210)(143,209)(144,208)
(145,207)(146,206)(147,205)(148,197)(149,203)(150,202)(151,201)(152,200)
(153,199)(154,198)(155,218)(156,224)(157,223)(158,222)(159,221)(160,220)
(161,219)(162,211)(163,217)(164,216)(165,215)(166,214)(167,213)(168,212);;
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, 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);;
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)
( 57,169)( 58,170)( 59,171)( 60,172)( 61,173)( 62,174)( 63,175)( 64,176)
( 65,177)( 66,178)( 67,179)( 68,180)( 69,181)( 70,182)( 71,183)( 72,184)
( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)( 80,192)
( 81,193)( 82,194)( 83,195)( 84,196)( 85,204)( 86,205)( 87,206)( 88,207)
( 89,208)( 90,209)( 91,210)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)
( 97,202)( 98,203)( 99,218)(100,219)(101,220)(102,221)(103,222)(104,223)
(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*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*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, 57)(  2, 63)(  3, 62)(  4, 61)(  5, 60)(  6, 59)(  7, 58)
(  8, 64)(  9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 71)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 78)( 23, 84)
( 24, 83)( 25, 82)( 26, 81)( 27, 80)( 28, 79)( 29, 92)( 30, 98)( 31, 97)
( 32, 96)( 33, 95)( 34, 94)( 35, 93)( 36, 85)( 37, 91)( 38, 90)( 39, 89)
( 40, 88)( 41, 87)( 42, 86)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)
( 48,108)( 49,107)( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)
( 56,100)(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)(141,204)(142,210)(143,209)
(144,208)(145,207)(146,206)(147,205)(148,197)(149,203)(150,202)(151,201)
(152,200)(153,199)(154,198)(155,218)(156,224)(157,223)(158,222)(159,221)
(160,220)(161,219)(162,211)(163,217)(164,216)(165,215)(166,214)(167,213)
(168,212);
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, 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);
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)( 57,169)( 58,170)( 59,171)( 60,172)( 61,173)( 62,174)( 63,175)
( 64,176)( 65,177)( 66,178)( 67,179)( 68,180)( 69,181)( 70,182)( 71,183)
( 72,184)( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)
( 80,192)( 81,193)( 82,194)( 83,195)( 84,196)( 85,204)( 86,205)( 87,206)
( 88,207)( 89,208)( 90,209)( 91,210)( 92,197)( 93,198)( 94,199)( 95,200)
( 96,201)( 97,202)( 98,203)( 99,218)(100,219)(101,220)(102,221)(103,222)
(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)
(112,217);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*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*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