Polytope of Type {4,8,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,14}*1792b
if this polytope has a name.
Group : SmallGroup(1792,323566)
Rank : 4
Schlafli Type : {4,8,14}
Number of vertices, edges, etc : 8, 32, 112, 14
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 : {4,4,14}*896
   4-fold quotients : {4,4,14}*448
   7-fold quotients : {4,8,2}*256b
   8-fold quotients : {2,4,14}*224, {4,2,14}*224
   14-fold quotients : {4,4,2}*128
   16-fold quotients : {4,2,7}*112, {2,2,14}*112
   28-fold quotients : {4,4,2}*64
   32-fold quotients : {2,2,7}*56
   56-fold quotients : {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 := ( 29, 43)( 30, 44)( 31, 45)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 50)
( 37, 51)( 38, 52)( 39, 53)( 40, 54)( 41, 55)( 42, 56)( 85, 99)( 86,100)
( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)( 94,108)
( 95,109)( 96,110)( 97,111)( 98,112)(113,127)(114,128)(115,129)(116,130)
(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)
(125,139)(126,140)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)
(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196);;
s1 := ( 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);;
s2 := (  1,197)(  2,203)(  3,202)(  4,201)(  5,200)(  6,199)(  7,198)(  8,204)
(  9,210)( 10,209)( 11,208)( 12,207)( 13,206)( 14,205)( 15,211)( 16,217)
( 17,216)( 18,215)( 19,214)( 20,213)( 21,212)( 22,218)( 23,224)( 24,223)
( 25,222)( 26,221)( 27,220)( 28,219)( 29,183)( 30,189)( 31,188)( 32,187)
( 33,186)( 34,185)( 35,184)( 36,190)( 37,196)( 38,195)( 39,194)( 40,193)
( 41,192)( 42,191)( 43,169)( 44,175)( 45,174)( 46,173)( 47,172)( 48,171)
( 49,170)( 50,176)( 51,182)( 52,181)( 53,180)( 54,179)( 55,178)( 56,177)
( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)( 64,148)
( 65,154)( 66,153)( 67,152)( 68,151)( 69,150)( 70,149)( 71,155)( 72,161)
( 73,160)( 74,159)( 75,158)( 76,157)( 77,156)( 78,162)( 79,168)( 80,167)
( 81,166)( 82,165)( 83,164)( 84,163)( 85,127)( 86,133)( 87,132)( 88,131)
( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)( 96,137)
( 97,136)( 98,135)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)
(105,114)(106,120)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121);;
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, 86)( 30, 85)( 31, 91)( 32, 90)
( 33, 89)( 34, 88)( 35, 87)( 36, 93)( 37, 92)( 38, 98)( 39, 97)( 40, 96)
( 41, 95)( 42, 94)( 43,100)( 44, 99)( 45,105)( 46,104)( 47,103)( 48,102)
( 49,101)( 50,107)( 51,106)( 52,112)( 53,111)( 54,110)( 55,109)( 56,108)
(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)
(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)
(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)
(137,195)(138,194)(139,193)(140,192)(141,198)(142,197)(143,203)(144,202)
(145,201)(146,200)(147,199)(148,205)(149,204)(150,210)(151,209)(152,208)
(153,207)(154,206)(155,212)(156,211)(157,217)(158,216)(159,215)(160,214)
(161,213)(162,219)(163,218)(164,224)(165,223)(166,222)(167,221)(168,220);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
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)!( 29, 43)( 30, 44)( 31, 45)( 32, 46)( 33, 47)( 34, 48)( 35, 49)
( 36, 50)( 37, 51)( 38, 52)( 39, 53)( 40, 54)( 41, 55)( 42, 56)( 85, 99)
( 86,100)( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)
( 94,108)( 95,109)( 96,110)( 97,111)( 98,112)(113,127)(114,128)(115,129)
(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)
(124,138)(125,139)(126,140)(169,183)(170,184)(171,185)(172,186)(173,187)
(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)
(182,196);
s1 := 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);
s2 := Sym(224)!(  1,197)(  2,203)(  3,202)(  4,201)(  5,200)(  6,199)(  7,198)
(  8,204)(  9,210)( 10,209)( 11,208)( 12,207)( 13,206)( 14,205)( 15,211)
( 16,217)( 17,216)( 18,215)( 19,214)( 20,213)( 21,212)( 22,218)( 23,224)
( 24,223)( 25,222)( 26,221)( 27,220)( 28,219)( 29,183)( 30,189)( 31,188)
( 32,187)( 33,186)( 34,185)( 35,184)( 36,190)( 37,196)( 38,195)( 39,194)
( 40,193)( 41,192)( 42,191)( 43,169)( 44,175)( 45,174)( 46,173)( 47,172)
( 48,171)( 49,170)( 50,176)( 51,182)( 52,181)( 53,180)( 54,179)( 55,178)
( 56,177)( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)
( 64,148)( 65,154)( 66,153)( 67,152)( 68,151)( 69,150)( 70,149)( 71,155)
( 72,161)( 73,160)( 74,159)( 75,158)( 76,157)( 77,156)( 78,162)( 79,168)
( 80,167)( 81,166)( 82,165)( 83,164)( 84,163)( 85,127)( 86,133)( 87,132)
( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)
( 96,137)( 97,136)( 98,135)( 99,113)(100,119)(101,118)(102,117)(103,116)
(104,115)(105,114)(106,120)(107,126)(108,125)(109,124)(110,123)(111,122)
(112,121);
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, 86)( 30, 85)( 31, 91)
( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 36, 93)( 37, 92)( 38, 98)( 39, 97)
( 40, 96)( 41, 95)( 42, 94)( 43,100)( 44, 99)( 45,105)( 46,104)( 47,103)
( 48,102)( 49,101)( 50,107)( 51,106)( 52,112)( 53,111)( 54,110)( 55,109)
( 56,108)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)
(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)
(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)
(136,196)(137,195)(138,194)(139,193)(140,192)(141,198)(142,197)(143,203)
(144,202)(145,201)(146,200)(147,199)(148,205)(149,204)(150,210)(151,209)
(152,208)(153,207)(154,206)(155,212)(156,211)(157,217)(158,216)(159,215)
(160,214)(161,213)(162,219)(163,218)(164,224)(165,223)(166,222)(167,221)
(168,220);
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
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