Polytope of Type {4,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,28}*1792
if this polytope has a name.
Group : SmallGroup(1792,90280)
Rank : 3
Schlafli Type : {4,28}
Number of vertices, edges, etc : 32, 448, 224
Order of s₀s₁s₂ : 56
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 : {4,28}*896
   4-fold quotients : {4,28}*448
   7-fold quotients : {4,4}*256
   8-fold quotients : {4,28}*224
   14-fold quotients : {4,4}*128
   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 := ( 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)( 57,106)( 58,107)
( 59,108)( 60,109)( 61,110)( 62,111)( 63,112)( 64, 99)( 65,100)( 66,101)
( 67,102)( 68,103)( 69,104)( 70,105)( 71, 85)( 72, 86)( 73, 87)( 74, 88)
( 75, 89)( 76, 90)( 77, 91)( 78, 92)( 79, 93)( 80, 94)( 81, 95)( 82, 96)
( 83, 97)( 84, 98)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)
(133,140)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)
(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)
(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)(183,197)(184,198)
(185,199)(186,200)(187,201)(188,202)(189,203)(190,204)(191,205)(192,206)
(193,207)(194,208)(195,209)(196,210);;
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 := (  1,114)(  2,113)(  3,119)(  4,118)(  5,117)(  6,116)(  7,115)(  8,121)
(  9,120)( 10,126)( 11,125)( 12,124)( 13,123)( 14,122)( 15,135)( 16,134)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,136)( 22,128)( 23,127)( 24,133)
( 25,132)( 26,131)( 27,130)( 28,129)( 29,149)( 30,148)( 31,154)( 32,153)
( 33,152)( 34,151)( 35,150)( 36,142)( 37,141)( 38,147)( 39,146)( 40,145)
( 41,144)( 42,143)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)( 56,164)
( 57,170)( 58,169)( 59,175)( 60,174)( 61,173)( 62,172)( 63,171)( 64,177)
( 65,176)( 66,182)( 67,181)( 68,180)( 69,179)( 70,178)( 71,191)( 72,190)
( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,184)( 79,183)( 80,189)
( 81,188)( 82,187)( 83,186)( 84,185)( 85,205)( 86,204)( 87,210)( 88,209)
( 89,208)( 90,207)( 91,206)( 92,198)( 93,197)( 94,203)( 95,202)( 96,201)
( 97,200)( 98,199)( 99,212)(100,211)(101,217)(102,216)(103,215)(104,214)
(105,213)(106,219)(107,218)(108,224)(109,223)(110,222)(111,221)(112,220);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*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)!( 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)( 57,106)
( 58,107)( 59,108)( 60,109)( 61,110)( 62,111)( 63,112)( 64, 99)( 65,100)
( 66,101)( 67,102)( 68,103)( 69,104)( 70,105)( 71, 85)( 72, 86)( 73, 87)
( 74, 88)( 75, 89)( 76, 90)( 77, 91)( 78, 92)( 79, 93)( 80, 94)( 81, 95)
( 82, 96)( 83, 97)( 84, 98)(127,134)(128,135)(129,136)(130,137)(131,138)
(132,139)(133,140)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)
(161,168)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)
(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)(183,197)
(184,198)(185,199)(186,200)(187,201)(188,202)(189,203)(190,204)(191,205)
(192,206)(193,207)(194,208)(195,209)(196,210);
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)!(  1,114)(  2,113)(  3,119)(  4,118)(  5,117)(  6,116)(  7,115)
(  8,121)(  9,120)( 10,126)( 11,125)( 12,124)( 13,123)( 14,122)( 15,135)
( 16,134)( 17,140)( 18,139)( 19,138)( 20,137)( 21,136)( 22,128)( 23,127)
( 24,133)( 25,132)( 26,131)( 27,130)( 28,129)( 29,149)( 30,148)( 31,154)
( 32,153)( 33,152)( 34,151)( 35,150)( 36,142)( 37,141)( 38,147)( 39,146)
( 40,145)( 41,144)( 42,143)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,163)( 51,162)( 52,168)( 53,167)( 54,166)( 55,165)
( 56,164)( 57,170)( 58,169)( 59,175)( 60,174)( 61,173)( 62,172)( 63,171)
( 64,177)( 65,176)( 66,182)( 67,181)( 68,180)( 69,179)( 70,178)( 71,191)
( 72,190)( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,184)( 79,183)
( 80,189)( 81,188)( 82,187)( 83,186)( 84,185)( 85,205)( 86,204)( 87,210)
( 88,209)( 89,208)( 90,207)( 91,206)( 92,198)( 93,197)( 94,203)( 95,202)
( 96,201)( 97,200)( 98,199)( 99,212)(100,211)(101,217)(102,216)(103,215)
(104,214)(105,213)(106,219)(107,218)(108,224)(109,223)(110,222)(111,221)
(112,220);
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*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