Questions?
See the FAQ
or other info.

Polytope of Type {6,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,16}*576
if this polytope has a name.
Group : SmallGroup(576,5158)
Rank : 3
Schlafli Type : {6,16}
Number of vertices, edges, etc : 18, 144, 48
Order of s0s1s2 : 16
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,16,2} of size 1152
Vertex Figure Of :
   {2,6,16} of size 1152
   {3,6,16} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,8}*288
   4-fold quotients : {6,4}*144
   8-fold quotients : {6,4}*72
   9-fold quotients : {2,16}*64
   18-fold quotients : {2,8}*32
   36-fold quotients : {2,4}*16
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,16}*1152a, {6,32}*1152
   3-fold covers : {6,16}*1728a, {6,48}*1728d, {6,48}*1728e, {6,16}*1728b, {6,48}*1728g, {6,48}*1728h
Permutation Representation (GAP) :
s0 := (  1,127)(  2,129)(  3,128)(  4,133)(  5,135)(  6,134)(  7,130)(  8,132)
(  9,131)( 10,136)( 11,138)( 12,137)( 13,142)( 14,144)( 15,143)( 16,139)
( 17,141)( 18,140)( 19,118)( 20,120)( 21,119)( 22,124)( 23,126)( 24,125)
( 25,121)( 26,123)( 27,122)( 28,109)( 29,111)( 30,110)( 31,115)( 32,117)
( 33,116)( 34,112)( 35,114)( 36,113)( 37, 73)( 38, 75)( 39, 74)( 40, 79)
( 41, 81)( 42, 80)( 43, 76)( 44, 78)( 45, 77)( 46, 82)( 47, 84)( 48, 83)
( 49, 88)( 50, 90)( 51, 89)( 52, 85)( 53, 87)( 54, 86)( 55, 91)( 56, 93)
( 57, 92)( 58, 97)( 59, 99)( 60, 98)( 61, 94)( 62, 96)( 63, 95)( 64,100)
( 65,102)( 66,101)( 67,106)( 68,108)( 69,107)( 70,103)( 71,105)( 72,104)
(145,271)(146,273)(147,272)(148,277)(149,279)(150,278)(151,274)(152,276)
(153,275)(154,280)(155,282)(156,281)(157,286)(158,288)(159,287)(160,283)
(161,285)(162,284)(163,262)(164,264)(165,263)(166,268)(167,270)(168,269)
(169,265)(170,267)(171,266)(172,253)(173,255)(174,254)(175,259)(176,261)
(177,260)(178,256)(179,258)(180,257)(181,217)(182,219)(183,218)(184,223)
(185,225)(186,224)(187,220)(188,222)(189,221)(190,226)(191,228)(192,227)
(193,232)(194,234)(195,233)(196,229)(197,231)(198,230)(199,235)(200,237)
(201,236)(202,241)(203,243)(204,242)(205,238)(206,240)(207,239)(208,244)
(209,246)(210,245)(211,250)(212,252)(213,251)(214,247)(215,249)(216,248);;
s1 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)( 20, 32)
( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)( 37, 58)
( 38, 59)( 39, 60)( 40, 55)( 41, 56)( 42, 57)( 43, 61)( 44, 62)( 45, 63)
( 46, 67)( 47, 68)( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)
( 54, 72)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)( 79, 97)
( 80, 98)( 81, 99)( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)( 87,102)
( 88,106)( 89,107)( 90,108)(109,121)(110,122)(111,123)(112,118)(113,119)
(114,120)(115,124)(116,125)(117,126)(127,130)(128,131)(129,132)(136,139)
(137,140)(138,141)(145,184)(146,185)(147,186)(148,181)(149,182)(150,183)
(151,187)(152,188)(153,189)(154,193)(155,194)(156,195)(157,190)(158,191)
(159,192)(160,196)(161,197)(162,198)(163,211)(164,212)(165,213)(166,208)
(167,209)(168,210)(169,214)(170,215)(171,216)(172,202)(173,203)(174,204)
(175,199)(176,200)(177,201)(178,205)(179,206)(180,207)(217,274)(218,275)
(219,276)(220,271)(221,272)(222,273)(223,277)(224,278)(225,279)(226,283)
(227,284)(228,285)(229,280)(230,281)(231,282)(232,286)(233,287)(234,288)
(235,256)(236,257)(237,258)(238,253)(239,254)(240,255)(241,259)(242,260)
(243,261)(244,265)(245,266)(246,267)(247,262)(248,263)(249,264)(250,268)
(251,269)(252,270);;
s2 := (  1,145)(  2,148)(  3,151)(  4,146)(  5,149)(  6,152)(  7,147)(  8,150)
(  9,153)( 10,154)( 11,157)( 12,160)( 13,155)( 14,158)( 15,161)( 16,156)
( 17,159)( 18,162)( 19,172)( 20,175)( 21,178)( 22,173)( 23,176)( 24,179)
( 25,174)( 26,177)( 27,180)( 28,163)( 29,166)( 30,169)( 31,164)( 32,167)
( 33,170)( 34,165)( 35,168)( 36,171)( 37,199)( 38,202)( 39,205)( 40,200)
( 41,203)( 42,206)( 43,201)( 44,204)( 45,207)( 46,208)( 47,211)( 48,214)
( 49,209)( 50,212)( 51,215)( 52,210)( 53,213)( 54,216)( 55,181)( 56,184)
( 57,187)( 58,182)( 59,185)( 60,188)( 61,183)( 62,186)( 63,189)( 64,190)
( 65,193)( 66,196)( 67,191)( 68,194)( 69,197)( 70,192)( 71,195)( 72,198)
( 73,235)( 74,238)( 75,241)( 76,236)( 77,239)( 78,242)( 79,237)( 80,240)
( 81,243)( 82,244)( 83,247)( 84,250)( 85,245)( 86,248)( 87,251)( 88,246)
( 89,249)( 90,252)( 91,217)( 92,220)( 93,223)( 94,218)( 95,221)( 96,224)
( 97,219)( 98,222)( 99,225)(100,226)(101,229)(102,232)(103,227)(104,230)
(105,233)(106,228)(107,231)(108,234)(109,262)(110,265)(111,268)(112,263)
(113,266)(114,269)(115,264)(116,267)(117,270)(118,253)(119,256)(120,259)
(121,254)(122,257)(123,260)(124,255)(125,258)(126,261)(127,271)(128,274)
(129,277)(130,272)(131,275)(132,278)(133,273)(134,276)(135,279)(136,280)
(137,283)(138,286)(139,281)(140,284)(141,287)(142,282)(143,285)(144,288);;
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*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(288)!(  1,127)(  2,129)(  3,128)(  4,133)(  5,135)(  6,134)(  7,130)
(  8,132)(  9,131)( 10,136)( 11,138)( 12,137)( 13,142)( 14,144)( 15,143)
( 16,139)( 17,141)( 18,140)( 19,118)( 20,120)( 21,119)( 22,124)( 23,126)
( 24,125)( 25,121)( 26,123)( 27,122)( 28,109)( 29,111)( 30,110)( 31,115)
( 32,117)( 33,116)( 34,112)( 35,114)( 36,113)( 37, 73)( 38, 75)( 39, 74)
( 40, 79)( 41, 81)( 42, 80)( 43, 76)( 44, 78)( 45, 77)( 46, 82)( 47, 84)
( 48, 83)( 49, 88)( 50, 90)( 51, 89)( 52, 85)( 53, 87)( 54, 86)( 55, 91)
( 56, 93)( 57, 92)( 58, 97)( 59, 99)( 60, 98)( 61, 94)( 62, 96)( 63, 95)
( 64,100)( 65,102)( 66,101)( 67,106)( 68,108)( 69,107)( 70,103)( 71,105)
( 72,104)(145,271)(146,273)(147,272)(148,277)(149,279)(150,278)(151,274)
(152,276)(153,275)(154,280)(155,282)(156,281)(157,286)(158,288)(159,287)
(160,283)(161,285)(162,284)(163,262)(164,264)(165,263)(166,268)(167,270)
(168,269)(169,265)(170,267)(171,266)(172,253)(173,255)(174,254)(175,259)
(176,261)(177,260)(178,256)(179,258)(180,257)(181,217)(182,219)(183,218)
(184,223)(185,225)(186,224)(187,220)(188,222)(189,221)(190,226)(191,228)
(192,227)(193,232)(194,234)(195,233)(196,229)(197,231)(198,230)(199,235)
(200,237)(201,236)(202,241)(203,243)(204,242)(205,238)(206,240)(207,239)
(208,244)(209,246)(210,245)(211,250)(212,252)(213,251)(214,247)(215,249)
(216,248);
s1 := Sym(288)!(  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)
( 20, 32)( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)
( 37, 58)( 38, 59)( 39, 60)( 40, 55)( 41, 56)( 42, 57)( 43, 61)( 44, 62)
( 45, 63)( 46, 67)( 47, 68)( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)
( 53, 71)( 54, 72)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)
( 79, 97)( 80, 98)( 81, 99)( 82,103)( 83,104)( 84,105)( 85,100)( 86,101)
( 87,102)( 88,106)( 89,107)( 90,108)(109,121)(110,122)(111,123)(112,118)
(113,119)(114,120)(115,124)(116,125)(117,126)(127,130)(128,131)(129,132)
(136,139)(137,140)(138,141)(145,184)(146,185)(147,186)(148,181)(149,182)
(150,183)(151,187)(152,188)(153,189)(154,193)(155,194)(156,195)(157,190)
(158,191)(159,192)(160,196)(161,197)(162,198)(163,211)(164,212)(165,213)
(166,208)(167,209)(168,210)(169,214)(170,215)(171,216)(172,202)(173,203)
(174,204)(175,199)(176,200)(177,201)(178,205)(179,206)(180,207)(217,274)
(218,275)(219,276)(220,271)(221,272)(222,273)(223,277)(224,278)(225,279)
(226,283)(227,284)(228,285)(229,280)(230,281)(231,282)(232,286)(233,287)
(234,288)(235,256)(236,257)(237,258)(238,253)(239,254)(240,255)(241,259)
(242,260)(243,261)(244,265)(245,266)(246,267)(247,262)(248,263)(249,264)
(250,268)(251,269)(252,270);
s2 := Sym(288)!(  1,145)(  2,148)(  3,151)(  4,146)(  5,149)(  6,152)(  7,147)
(  8,150)(  9,153)( 10,154)( 11,157)( 12,160)( 13,155)( 14,158)( 15,161)
( 16,156)( 17,159)( 18,162)( 19,172)( 20,175)( 21,178)( 22,173)( 23,176)
( 24,179)( 25,174)( 26,177)( 27,180)( 28,163)( 29,166)( 30,169)( 31,164)
( 32,167)( 33,170)( 34,165)( 35,168)( 36,171)( 37,199)( 38,202)( 39,205)
( 40,200)( 41,203)( 42,206)( 43,201)( 44,204)( 45,207)( 46,208)( 47,211)
( 48,214)( 49,209)( 50,212)( 51,215)( 52,210)( 53,213)( 54,216)( 55,181)
( 56,184)( 57,187)( 58,182)( 59,185)( 60,188)( 61,183)( 62,186)( 63,189)
( 64,190)( 65,193)( 66,196)( 67,191)( 68,194)( 69,197)( 70,192)( 71,195)
( 72,198)( 73,235)( 74,238)( 75,241)( 76,236)( 77,239)( 78,242)( 79,237)
( 80,240)( 81,243)( 82,244)( 83,247)( 84,250)( 85,245)( 86,248)( 87,251)
( 88,246)( 89,249)( 90,252)( 91,217)( 92,220)( 93,223)( 94,218)( 95,221)
( 96,224)( 97,219)( 98,222)( 99,225)(100,226)(101,229)(102,232)(103,227)
(104,230)(105,233)(106,228)(107,231)(108,234)(109,262)(110,265)(111,268)
(112,263)(113,266)(114,269)(115,264)(116,267)(117,270)(118,253)(119,256)
(120,259)(121,254)(122,257)(123,260)(124,255)(125,258)(126,261)(127,271)
(128,274)(129,277)(130,272)(131,275)(132,278)(133,273)(134,276)(135,279)
(136,280)(137,283)(138,286)(139,281)(140,284)(141,287)(142,282)(143,285)
(144,288);
poly := sub<Sym(288)|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*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*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 >; 
 
References : None.
to this polytope