Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*1800
Also Known As : {4,4}_(15,0), {4,4|15}. if this polytope has another name.
Group : SmallGroup(1800,664)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 225, 450, 225
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 15
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {4,4}*200
   25-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)( 11, 16)
( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)( 33, 43)
( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 52, 62)( 53, 73)( 54, 59)( 55, 70)
( 56, 71)( 58, 68)( 60, 65)( 61, 66)( 64, 74)( 67, 72)( 76,151)( 77,162)
( 78,173)( 79,159)( 80,170)( 81,171)( 82,157)( 83,168)( 84,154)( 85,165)
( 86,166)( 87,152)( 88,163)( 89,174)( 90,160)( 91,161)( 92,172)( 93,158)
( 94,169)( 95,155)( 96,156)( 97,167)( 98,153)( 99,164)(100,175)(101,176)
(102,187)(103,198)(104,184)(105,195)(106,196)(107,182)(108,193)(109,179)
(110,190)(111,191)(112,177)(113,188)(114,199)(115,185)(116,186)(117,197)
(118,183)(119,194)(120,180)(121,181)(122,192)(123,178)(124,189)(125,200)
(126,201)(127,212)(128,223)(129,209)(130,220)(131,221)(132,207)(133,218)
(134,204)(135,215)(136,216)(137,202)(138,213)(139,224)(140,210)(141,211)
(142,222)(143,208)(144,219)(145,205)(146,206)(147,217)(148,203)(149,214)
(150,225);;
s1 := (  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)
( 15, 17)( 19, 21)( 26,151)( 27,159)( 28,162)( 29,170)( 30,173)( 31,163)
( 32,166)( 33,174)( 34,152)( 35,160)( 36,175)( 37,153)( 38,156)( 39,164)
( 40,167)( 41,157)( 42,165)( 43,168)( 44,171)( 45,154)( 46,169)( 47,172)
( 48,155)( 49,158)( 50,161)( 51, 76)( 52, 84)( 53, 87)( 54, 95)( 55, 98)
( 56, 88)( 57, 91)( 58, 99)( 59, 77)( 60, 85)( 61,100)( 62, 78)( 63, 81)
( 64, 89)( 65, 92)( 66, 82)( 67, 90)( 68, 93)( 69, 96)( 70, 79)( 71, 94)
( 72, 97)( 73, 80)( 74, 83)( 75, 86)(101,201)(102,209)(103,212)(104,220)
(105,223)(106,213)(107,216)(108,224)(109,202)(110,210)(111,225)(112,203)
(113,206)(114,214)(115,217)(116,207)(117,215)(118,218)(119,221)(120,204)
(121,219)(122,222)(123,205)(124,208)(125,211)(127,134)(128,137)(129,145)
(130,148)(131,138)(132,141)(133,149)(136,150)(140,142)(144,146)(177,184)
(178,187)(179,195)(180,198)(181,188)(182,191)(183,199)(186,200)(190,192)
(194,196);;
s2 := (  1, 32)(  2, 46)(  3, 40)(  4, 29)(  5, 43)(  6, 37)(  7, 26)(  8, 45)
(  9, 34)( 10, 48)( 11, 42)( 12, 31)( 13, 50)( 14, 39)( 15, 28)( 16, 47)
( 17, 36)( 18, 30)( 19, 44)( 20, 33)( 21, 27)( 22, 41)( 23, 35)( 24, 49)
( 25, 38)( 51, 57)( 52, 71)( 53, 65)( 55, 68)( 56, 62)( 58, 70)( 60, 73)
( 61, 67)( 63, 75)( 66, 72)( 76,107)( 77,121)( 78,115)( 79,104)( 80,118)
( 81,112)( 82,101)( 83,120)( 84,109)( 85,123)( 86,117)( 87,106)( 88,125)
( 89,114)( 90,103)( 91,122)( 92,111)( 93,105)( 94,119)( 95,108)( 96,102)
( 97,116)( 98,110)( 99,124)(100,113)(126,132)(127,146)(128,140)(130,143)
(131,137)(133,145)(135,148)(136,142)(138,150)(141,147)(151,182)(152,196)
(153,190)(154,179)(155,193)(156,187)(157,176)(158,195)(159,184)(160,198)
(161,192)(162,181)(163,200)(164,189)(165,178)(166,197)(167,186)(168,180)
(169,194)(170,183)(171,177)(172,191)(173,185)(174,199)(175,188)(201,207)
(202,221)(203,215)(205,218)(206,212)(208,220)(210,223)(211,217)(213,225)
(216,222);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(225)!(  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)
( 11, 16)( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)
( 33, 43)( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 52, 62)( 53, 73)( 54, 59)
( 55, 70)( 56, 71)( 58, 68)( 60, 65)( 61, 66)( 64, 74)( 67, 72)( 76,151)
( 77,162)( 78,173)( 79,159)( 80,170)( 81,171)( 82,157)( 83,168)( 84,154)
( 85,165)( 86,166)( 87,152)( 88,163)( 89,174)( 90,160)( 91,161)( 92,172)
( 93,158)( 94,169)( 95,155)( 96,156)( 97,167)( 98,153)( 99,164)(100,175)
(101,176)(102,187)(103,198)(104,184)(105,195)(106,196)(107,182)(108,193)
(109,179)(110,190)(111,191)(112,177)(113,188)(114,199)(115,185)(116,186)
(117,197)(118,183)(119,194)(120,180)(121,181)(122,192)(123,178)(124,189)
(125,200)(126,201)(127,212)(128,223)(129,209)(130,220)(131,221)(132,207)
(133,218)(134,204)(135,215)(136,216)(137,202)(138,213)(139,224)(140,210)
(141,211)(142,222)(143,208)(144,219)(145,205)(146,206)(147,217)(148,203)
(149,214)(150,225);
s1 := Sym(225)!(  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)
( 11, 25)( 15, 17)( 19, 21)( 26,151)( 27,159)( 28,162)( 29,170)( 30,173)
( 31,163)( 32,166)( 33,174)( 34,152)( 35,160)( 36,175)( 37,153)( 38,156)
( 39,164)( 40,167)( 41,157)( 42,165)( 43,168)( 44,171)( 45,154)( 46,169)
( 47,172)( 48,155)( 49,158)( 50,161)( 51, 76)( 52, 84)( 53, 87)( 54, 95)
( 55, 98)( 56, 88)( 57, 91)( 58, 99)( 59, 77)( 60, 85)( 61,100)( 62, 78)
( 63, 81)( 64, 89)( 65, 92)( 66, 82)( 67, 90)( 68, 93)( 69, 96)( 70, 79)
( 71, 94)( 72, 97)( 73, 80)( 74, 83)( 75, 86)(101,201)(102,209)(103,212)
(104,220)(105,223)(106,213)(107,216)(108,224)(109,202)(110,210)(111,225)
(112,203)(113,206)(114,214)(115,217)(116,207)(117,215)(118,218)(119,221)
(120,204)(121,219)(122,222)(123,205)(124,208)(125,211)(127,134)(128,137)
(129,145)(130,148)(131,138)(132,141)(133,149)(136,150)(140,142)(144,146)
(177,184)(178,187)(179,195)(180,198)(181,188)(182,191)(183,199)(186,200)
(190,192)(194,196);
s2 := Sym(225)!(  1, 32)(  2, 46)(  3, 40)(  4, 29)(  5, 43)(  6, 37)(  7, 26)
(  8, 45)(  9, 34)( 10, 48)( 11, 42)( 12, 31)( 13, 50)( 14, 39)( 15, 28)
( 16, 47)( 17, 36)( 18, 30)( 19, 44)( 20, 33)( 21, 27)( 22, 41)( 23, 35)
( 24, 49)( 25, 38)( 51, 57)( 52, 71)( 53, 65)( 55, 68)( 56, 62)( 58, 70)
( 60, 73)( 61, 67)( 63, 75)( 66, 72)( 76,107)( 77,121)( 78,115)( 79,104)
( 80,118)( 81,112)( 82,101)( 83,120)( 84,109)( 85,123)( 86,117)( 87,106)
( 88,125)( 89,114)( 90,103)( 91,122)( 92,111)( 93,105)( 94,119)( 95,108)
( 96,102)( 97,116)( 98,110)( 99,124)(100,113)(126,132)(127,146)(128,140)
(130,143)(131,137)(133,145)(135,148)(136,142)(138,150)(141,147)(151,182)
(152,196)(153,190)(154,179)(155,193)(156,187)(157,176)(158,195)(159,184)
(160,198)(161,192)(162,181)(163,200)(164,189)(165,178)(166,197)(167,186)
(168,180)(169,194)(170,183)(171,177)(172,191)(173,185)(174,199)(175,188)
(201,207)(202,221)(203,215)(205,218)(206,212)(208,220)(210,223)(211,217)
(213,225)(216,222);
poly := sub<Sym(225)|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, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope