Polytope of Type {18,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,28}*1008a
Also Known As : {18,28|2}. if this polytope has another name.
Group : SmallGroup(1008,158)
Rank : 3
Schlafli Type : {18,28}
Number of vertices, edges, etc : 18, 252, 28
Order of s₀s₁s₂ : 252
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {18,14}*504
   3-fold quotients : {6,28}*336a
   6-fold quotients : {6,14}*168
   7-fold quotients : {18,4}*144a
   9-fold quotients : {2,28}*112
   14-fold quotients : {18,2}*72
   18-fold quotients : {2,14}*56
   21-fold quotients : {6,4}*48a
   28-fold quotients : {9,2}*36
   36-fold quotients : {2,7}*28
   42-fold quotients : {6,2}*24
   63-fold quotients : {2,4}*16
   84-fold quotients : {3,2}*12
   126-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 22, 45)
( 23, 44)( 24, 43)( 25, 48)( 26, 47)( 27, 46)( 28, 51)( 29, 50)( 30, 49)
( 31, 54)( 32, 53)( 33, 52)( 34, 57)( 35, 56)( 36, 55)( 37, 60)( 38, 59)
( 39, 58)( 40, 63)( 41, 62)( 42, 61)( 65, 66)( 68, 69)( 71, 72)( 74, 75)
( 77, 78)( 80, 81)( 83, 84)( 85,108)( 86,107)( 87,106)( 88,111)( 89,110)
( 90,109)( 91,114)( 92,113)( 93,112)( 94,117)( 95,116)( 96,115)( 97,120)
( 98,119)( 99,118)(100,123)(101,122)(102,121)(103,126)(104,125)(105,124)
(128,129)(131,132)(134,135)(137,138)(140,141)(143,144)(146,147)(148,171)
(149,170)(150,169)(151,174)(152,173)(153,172)(154,177)(155,176)(156,175)
(157,180)(158,179)(159,178)(160,183)(161,182)(162,181)(163,186)(164,185)
(165,184)(166,189)(167,188)(168,187)(191,192)(194,195)(197,198)(200,201)
(203,204)(206,207)(209,210)(211,234)(212,233)(213,232)(214,237)(215,236)
(216,235)(217,240)(218,239)(219,238)(220,243)(221,242)(222,241)(223,246)
(224,245)(225,244)(226,249)(227,248)(228,247)(229,252)(230,251)(231,250);;
s1 := (  1, 22)(  2, 24)(  3, 23)(  4, 40)(  5, 42)(  6, 41)(  7, 37)(  8, 39)
(  9, 38)( 10, 34)( 11, 36)( 12, 35)( 13, 31)( 14, 33)( 15, 32)( 16, 28)
( 17, 30)( 18, 29)( 19, 25)( 20, 27)( 21, 26)( 43, 45)( 46, 63)( 47, 62)
( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)( 64, 85)
( 65, 87)( 66, 86)( 67,103)( 68,105)( 69,104)( 70,100)( 71,102)( 72,101)
( 73, 97)( 74, 99)( 75, 98)( 76, 94)( 77, 96)( 78, 95)( 79, 91)( 80, 93)
( 81, 92)( 82, 88)( 83, 90)( 84, 89)(106,108)(109,126)(110,125)(111,124)
(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(127,211)(128,213)
(129,212)(130,229)(131,231)(132,230)(133,226)(134,228)(135,227)(136,223)
(137,225)(138,224)(139,220)(140,222)(141,221)(142,217)(143,219)(144,218)
(145,214)(146,216)(147,215)(148,190)(149,192)(150,191)(151,208)(152,210)
(153,209)(154,205)(155,207)(156,206)(157,202)(158,204)(159,203)(160,199)
(161,201)(162,200)(163,196)(164,198)(165,197)(166,193)(167,195)(168,194)
(169,234)(170,233)(171,232)(172,252)(173,251)(174,250)(175,249)(176,248)
(177,247)(178,246)(179,245)(180,244)(181,243)(182,242)(183,241)(184,240)
(185,239)(186,238)(187,237)(188,236)(189,235);;
s2 := (  1,130)(  2,131)(  3,132)(  4,127)(  5,128)(  6,129)(  7,145)(  8,146)
(  9,147)( 10,142)( 11,143)( 12,144)( 13,139)( 14,140)( 15,141)( 16,136)
( 17,137)( 18,138)( 19,133)( 20,134)( 21,135)( 22,151)( 23,152)( 24,153)
( 25,148)( 26,149)( 27,150)( 28,166)( 29,167)( 30,168)( 31,163)( 32,164)
( 33,165)( 34,160)( 35,161)( 36,162)( 37,157)( 38,158)( 39,159)( 40,154)
( 41,155)( 42,156)( 43,172)( 44,173)( 45,174)( 46,169)( 47,170)( 48,171)
( 49,187)( 50,188)( 51,189)( 52,184)( 53,185)( 54,186)( 55,181)( 56,182)
( 57,183)( 58,178)( 59,179)( 60,180)( 61,175)( 62,176)( 63,177)( 64,193)
( 65,194)( 66,195)( 67,190)( 68,191)( 69,192)( 70,208)( 71,209)( 72,210)
( 73,205)( 74,206)( 75,207)( 76,202)( 77,203)( 78,204)( 79,199)( 80,200)
( 81,201)( 82,196)( 83,197)( 84,198)( 85,214)( 86,215)( 87,216)( 88,211)
( 89,212)( 90,213)( 91,229)( 92,230)( 93,231)( 94,226)( 95,227)( 96,228)
( 97,223)( 98,224)( 99,225)(100,220)(101,221)(102,222)(103,217)(104,218)
(105,219)(106,235)(107,236)(108,237)(109,232)(110,233)(111,234)(112,250)
(113,251)(114,252)(115,247)(116,248)(117,249)(118,244)(119,245)(120,246)
(121,241)(122,242)(123,243)(124,238)(125,239)(126,240);;
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*s2*s1*s0*s1*s2*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*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(252)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 22, 45)( 23, 44)( 24, 43)( 25, 48)( 26, 47)( 27, 46)( 28, 51)( 29, 50)
( 30, 49)( 31, 54)( 32, 53)( 33, 52)( 34, 57)( 35, 56)( 36, 55)( 37, 60)
( 38, 59)( 39, 58)( 40, 63)( 41, 62)( 42, 61)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 85,108)( 86,107)( 87,106)( 88,111)
( 89,110)( 90,109)( 91,114)( 92,113)( 93,112)( 94,117)( 95,116)( 96,115)
( 97,120)( 98,119)( 99,118)(100,123)(101,122)(102,121)(103,126)(104,125)
(105,124)(128,129)(131,132)(134,135)(137,138)(140,141)(143,144)(146,147)
(148,171)(149,170)(150,169)(151,174)(152,173)(153,172)(154,177)(155,176)
(156,175)(157,180)(158,179)(159,178)(160,183)(161,182)(162,181)(163,186)
(164,185)(165,184)(166,189)(167,188)(168,187)(191,192)(194,195)(197,198)
(200,201)(203,204)(206,207)(209,210)(211,234)(212,233)(213,232)(214,237)
(215,236)(216,235)(217,240)(218,239)(219,238)(220,243)(221,242)(222,241)
(223,246)(224,245)(225,244)(226,249)(227,248)(228,247)(229,252)(230,251)
(231,250);
s1 := Sym(252)!(  1, 22)(  2, 24)(  3, 23)(  4, 40)(  5, 42)(  6, 41)(  7, 37)
(  8, 39)(  9, 38)( 10, 34)( 11, 36)( 12, 35)( 13, 31)( 14, 33)( 15, 32)
( 16, 28)( 17, 30)( 18, 29)( 19, 25)( 20, 27)( 21, 26)( 43, 45)( 46, 63)
( 47, 62)( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)
( 64, 85)( 65, 87)( 66, 86)( 67,103)( 68,105)( 69,104)( 70,100)( 71,102)
( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76, 94)( 77, 96)( 78, 95)( 79, 91)
( 80, 93)( 81, 92)( 82, 88)( 83, 90)( 84, 89)(106,108)(109,126)(110,125)
(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(127,211)
(128,213)(129,212)(130,229)(131,231)(132,230)(133,226)(134,228)(135,227)
(136,223)(137,225)(138,224)(139,220)(140,222)(141,221)(142,217)(143,219)
(144,218)(145,214)(146,216)(147,215)(148,190)(149,192)(150,191)(151,208)
(152,210)(153,209)(154,205)(155,207)(156,206)(157,202)(158,204)(159,203)
(160,199)(161,201)(162,200)(163,196)(164,198)(165,197)(166,193)(167,195)
(168,194)(169,234)(170,233)(171,232)(172,252)(173,251)(174,250)(175,249)
(176,248)(177,247)(178,246)(179,245)(180,244)(181,243)(182,242)(183,241)
(184,240)(185,239)(186,238)(187,237)(188,236)(189,235);
s2 := Sym(252)!(  1,130)(  2,131)(  3,132)(  4,127)(  5,128)(  6,129)(  7,145)
(  8,146)(  9,147)( 10,142)( 11,143)( 12,144)( 13,139)( 14,140)( 15,141)
( 16,136)( 17,137)( 18,138)( 19,133)( 20,134)( 21,135)( 22,151)( 23,152)
( 24,153)( 25,148)( 26,149)( 27,150)( 28,166)( 29,167)( 30,168)( 31,163)
( 32,164)( 33,165)( 34,160)( 35,161)( 36,162)( 37,157)( 38,158)( 39,159)
( 40,154)( 41,155)( 42,156)( 43,172)( 44,173)( 45,174)( 46,169)( 47,170)
( 48,171)( 49,187)( 50,188)( 51,189)( 52,184)( 53,185)( 54,186)( 55,181)
( 56,182)( 57,183)( 58,178)( 59,179)( 60,180)( 61,175)( 62,176)( 63,177)
( 64,193)( 65,194)( 66,195)( 67,190)( 68,191)( 69,192)( 70,208)( 71,209)
( 72,210)( 73,205)( 74,206)( 75,207)( 76,202)( 77,203)( 78,204)( 79,199)
( 80,200)( 81,201)( 82,196)( 83,197)( 84,198)( 85,214)( 86,215)( 87,216)
( 88,211)( 89,212)( 90,213)( 91,229)( 92,230)( 93,231)( 94,226)( 95,227)
( 96,228)( 97,223)( 98,224)( 99,225)(100,220)(101,221)(102,222)(103,217)
(104,218)(105,219)(106,235)(107,236)(108,237)(109,232)(110,233)(111,234)
(112,250)(113,251)(114,252)(115,247)(116,248)(117,249)(118,244)(119,245)
(120,246)(121,241)(122,242)(123,243)(124,238)(125,239)(126,240);
poly := sub<Sym(252)|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*s2*s1*s0*s1*s2*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*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