Polytope of Type {8,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,28}*1792b
if this polytope has a name.
Group : SmallGroup(1792,90281)
Rank : 3
Schlafli Type : {8,28}
Number of vertices, edges, etc : 32, 448, 112
Order of s₀s₁s₂ : 28
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {8,28}*896b
   4-fold quotients : {4,28}*448
   7-fold quotients : {8,4}*256b
   8-fold quotients : {4,28}*224
   14-fold quotients : {8,4}*128b
   16-fold quotients : {2,28}*112, {4,14}*112
   28-fold quotients : {4,4}*64
   32-fold quotients : {2,14}*56
   56-fold quotients : {4,4}*32
   64-fold quotients : {2,7}*28
   112-fold quotients : {2,4}*16, {4,2}*16
   224-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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