Polytope of Type {4,8,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,28}*1792a
if this polytope has a name.
Group : SmallGroup(1792,201129)
Rank : 4
Schlafli Type : {4,8,28}
Number of vertices, edges, etc : 4, 16, 112, 28
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 : {4,4,28}*896
   4-fold quotients : {2,4,28}*448, {4,2,28}*448, {4,4,14}*448
   7-fold quotients : {4,8,4}*256a
   8-fold quotients : {2,2,28}*224, {2,4,14}*224, {4,2,14}*224
   14-fold quotients : {4,4,4}*128
   16-fold quotients : {4,2,7}*112, {2,2,14}*112
   28-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   32-fold quotients : {2,2,7}*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 := ( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85, 92)( 86, 93)( 87, 94)
( 88, 95)( 89, 96)( 90, 97)( 91, 98)(113,141)(114,142)(115,143)(116,144)
(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)
(125,153)(126,154)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)
(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)
(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)
(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,218)(184,219)
(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)
(193,214)(194,215)(195,216)(196,217);;
s2 := (  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);;
s3 := (  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)(  8, 65)
(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)( 16, 71)
( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)( 24, 84)
( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 93)( 30, 92)( 31, 98)( 32, 97)
( 33, 96)( 34, 95)( 35, 94)( 36, 86)( 37, 85)( 38, 91)( 39, 90)( 40, 89)
( 41, 88)( 42, 87)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,108)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)
(113,184)(114,183)(115,189)(116,188)(117,187)(118,186)(119,185)(120,191)
(121,190)(122,196)(123,195)(124,194)(125,193)(126,192)(127,170)(128,169)
(129,175)(130,174)(131,173)(132,172)(133,171)(134,177)(135,176)(136,182)
(137,181)(138,180)(139,179)(140,178)(141,219)(142,218)(143,224)(144,223)
(145,222)(146,221)(147,220)(148,212)(149,211)(150,217)(151,216)(152,215)
(153,214)(154,213)(155,205)(156,204)(157,210)(158,209)(159,208)(160,207)
(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)(167,200)(168,199);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)!( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 71, 78)
( 72, 79)( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85, 92)( 86, 93)
( 87, 94)( 88, 95)( 89, 96)( 90, 97)( 91, 98)(113,141)(114,142)(115,143)
(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)
(124,152)(125,153)(126,154)(127,162)(128,163)(129,164)(130,165)(131,166)
(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)
(140,161)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)
(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,218)
(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)
(192,213)(193,214)(194,215)(195,216)(196,217);
s2 := 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);
s3 := Sym(224)!(  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)
(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)
( 16, 71)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)
( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 93)( 30, 92)( 31, 98)
( 32, 97)( 33, 96)( 34, 95)( 35, 94)( 36, 86)( 37, 85)( 38, 91)( 39, 90)
( 40, 89)( 41, 88)( 42, 87)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,108)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)
( 56,101)(113,184)(114,183)(115,189)(116,188)(117,187)(118,186)(119,185)
(120,191)(121,190)(122,196)(123,195)(124,194)(125,193)(126,192)(127,170)
(128,169)(129,175)(130,174)(131,173)(132,172)(133,171)(134,177)(135,176)
(136,182)(137,181)(138,180)(139,179)(140,178)(141,219)(142,218)(143,224)
(144,223)(145,222)(146,221)(147,220)(148,212)(149,211)(150,217)(151,216)
(152,215)(153,214)(154,213)(155,205)(156,204)(157,210)(158,209)(159,208)
(160,207)(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)(167,200)
(168,199);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope