Polytope of Type {14,8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,8,4}*1792b
if this polytope has a name.
Group : SmallGroup(1792,323566)
Rank : 4
Schlafli Type : {14,8,4}
Number of vertices, edges, etc : 14, 112, 32, 8
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 : {14,4,4}*896
   4-fold quotients : {14,4,4}*448
   7-fold quotients : {2,8,4}*256b
   8-fold quotients : {14,2,4}*224, {14,4,2}*224
   14-fold quotients : {2,4,4}*128
   16-fold quotients : {7,2,4}*112, {14,2,2}*112
   28-fold quotients : {2,4,4}*64
   32-fold quotients : {7,2,2}*56
   56-fold quotients : {2,2,4}*32, {2,4,2}*32
   112-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s3*s2*s1*s2*s3*s2> of order 2.
      4 facets:
         4 of {14,8}*224
      14 vertex figures:
         14 of 2-fold non-regular quotient of {8,4}*128b

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