Polytope of Type {8,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*576b
if this polytope has a name.
Group : SmallGroup(576,5410)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 24, 144, 36
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,12,2} of size 1152
Vertex Figure Of :
   {2,8,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*288
   4-fold quotients : {4,6}*144
   8-fold quotients : {4,6}*72
   9-fold quotients : {8,4}*64b
   18-fold quotients : {4,4}*32
   36-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,12}*1152a, {8,24}*1152a, {8,24}*1152d
   3-fold covers : {8,12}*1728d, {24,12}*1728m, {24,12}*1728n, {8,12}*1728h, {24,12}*1728t, {24,12}*1728x
Permutation Representation (GAP) :

s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)( 20, 29)
( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)
( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)
( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)(109,118)(110,119)(111,120)
(112,124)(113,125)(114,126)(115,121)(116,122)(117,123)(130,133)(131,134)
(132,135)(139,142)(140,143)(141,144)(145,181)(146,182)(147,183)(148,187)
(149,188)(150,189)(151,184)(152,185)(153,186)(154,190)(155,191)(156,192)
(157,196)(158,197)(159,198)(160,193)(161,194)(162,195)(163,208)(164,209)
(165,210)(166,214)(167,215)(168,216)(169,211)(170,212)(171,213)(172,199)
(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)(180,204)
(217,262)(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)(224,266)
(225,267)(226,253)(227,254)(228,255)(229,259)(230,260)(231,261)(232,256)
(233,257)(234,258)(235,271)(236,272)(237,273)(238,277)(239,278)(240,279)
(241,274)(242,275)(243,276)(244,280)(245,281)(246,282)(247,286)(248,287)
(249,288)(250,283)(251,284)(252,285);;
s1 := (  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)( 19,163)( 20,166)( 21,169)( 22,164)( 23,167)( 24,170)
( 25,165)( 26,168)( 27,171)( 28,172)( 29,175)( 30,178)( 31,173)( 32,176)
( 33,179)( 34,174)( 35,177)( 36,180)( 37,190)( 38,193)( 39,196)( 40,191)
( 41,194)( 42,197)( 43,192)( 44,195)( 45,198)( 46,181)( 47,184)( 48,187)
( 49,182)( 50,185)( 51,188)( 52,183)( 53,186)( 54,189)( 55,208)( 56,211)
( 57,214)( 58,209)( 59,212)( 60,215)( 61,210)( 62,213)( 63,216)( 64,199)
( 65,202)( 66,205)( 67,200)( 68,203)( 69,206)( 70,201)( 71,204)( 72,207)
( 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)(109,280)(110,283)(111,286)(112,281)
(113,284)(114,287)(115,282)(116,285)(117,288)(118,271)(119,274)(120,277)
(121,272)(122,275)(123,278)(124,273)(125,276)(126,279)(127,262)(128,265)
(129,268)(130,263)(131,266)(132,269)(133,264)(134,267)(135,270)(136,253)
(137,256)(138,259)(139,254)(140,257)(141,260)(142,255)(143,258)(144,261);;
s2 := (  1,128)(  2,127)(  3,129)(  4,134)(  5,133)(  6,135)(  7,131)(  8,130)
(  9,132)( 10,137)( 11,136)( 12,138)( 13,143)( 14,142)( 15,144)( 16,140)
( 17,139)( 18,141)( 19,110)( 20,109)( 21,111)( 22,116)( 23,115)( 24,117)
( 25,113)( 26,112)( 27,114)( 28,119)( 29,118)( 30,120)( 31,125)( 32,124)
( 33,126)( 34,122)( 35,121)( 36,123)( 37, 92)( 38, 91)( 39, 93)( 40, 98)
( 41, 97)( 42, 99)( 43, 95)( 44, 94)( 45, 96)( 46,101)( 47,100)( 48,102)
( 49,107)( 50,106)( 51,108)( 52,104)( 53,103)( 54,105)( 55, 74)( 56, 73)
( 57, 75)( 58, 80)( 59, 79)( 60, 81)( 61, 77)( 62, 76)( 63, 78)( 64, 83)
( 65, 82)( 66, 84)( 67, 89)( 68, 88)( 69, 90)( 70, 86)( 71, 85)( 72, 87)
(145,272)(146,271)(147,273)(148,278)(149,277)(150,279)(151,275)(152,274)
(153,276)(154,281)(155,280)(156,282)(157,287)(158,286)(159,288)(160,284)
(161,283)(162,285)(163,254)(164,253)(165,255)(166,260)(167,259)(168,261)
(169,257)(170,256)(171,258)(172,263)(173,262)(174,264)(175,269)(176,268)
(177,270)(178,266)(179,265)(180,267)(181,236)(182,235)(183,237)(184,242)
(185,241)(186,243)(187,239)(188,238)(189,240)(190,245)(191,244)(192,246)
(193,251)(194,250)(195,252)(196,248)(197,247)(198,249)(199,218)(200,217)
(201,219)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)(208,227)
(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)(216,231);;
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*s2*s0*s1*s2*s0*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, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(288)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)
( 20, 29)( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)
( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)
( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)(109,118)(110,119)
(111,120)(112,124)(113,125)(114,126)(115,121)(116,122)(117,123)(130,133)
(131,134)(132,135)(139,142)(140,143)(141,144)(145,181)(146,182)(147,183)
(148,187)(149,188)(150,189)(151,184)(152,185)(153,186)(154,190)(155,191)
(156,192)(157,196)(158,197)(159,198)(160,193)(161,194)(162,195)(163,208)
(164,209)(165,210)(166,214)(167,215)(168,216)(169,211)(170,212)(171,213)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(217,262)(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)
(224,266)(225,267)(226,253)(227,254)(228,255)(229,259)(230,260)(231,261)
(232,256)(233,257)(234,258)(235,271)(236,272)(237,273)(238,277)(239,278)
(240,279)(241,274)(242,275)(243,276)(244,280)(245,281)(246,282)(247,286)
(248,287)(249,288)(250,283)(251,284)(252,285);
s1 := 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)( 19,163)( 20,166)( 21,169)( 22,164)( 23,167)
( 24,170)( 25,165)( 26,168)( 27,171)( 28,172)( 29,175)( 30,178)( 31,173)
( 32,176)( 33,179)( 34,174)( 35,177)( 36,180)( 37,190)( 38,193)( 39,196)
( 40,191)( 41,194)( 42,197)( 43,192)( 44,195)( 45,198)( 46,181)( 47,184)
( 48,187)( 49,182)( 50,185)( 51,188)( 52,183)( 53,186)( 54,189)( 55,208)
( 56,211)( 57,214)( 58,209)( 59,212)( 60,215)( 61,210)( 62,213)( 63,216)
( 64,199)( 65,202)( 66,205)( 67,200)( 68,203)( 69,206)( 70,201)( 71,204)
( 72,207)( 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)(109,280)(110,283)(111,286)
(112,281)(113,284)(114,287)(115,282)(116,285)(117,288)(118,271)(119,274)
(120,277)(121,272)(122,275)(123,278)(124,273)(125,276)(126,279)(127,262)
(128,265)(129,268)(130,263)(131,266)(132,269)(133,264)(134,267)(135,270)
(136,253)(137,256)(138,259)(139,254)(140,257)(141,260)(142,255)(143,258)
(144,261);
s2 := Sym(288)!(  1,128)(  2,127)(  3,129)(  4,134)(  5,133)(  6,135)(  7,131)
(  8,130)(  9,132)( 10,137)( 11,136)( 12,138)( 13,143)( 14,142)( 15,144)
( 16,140)( 17,139)( 18,141)( 19,110)( 20,109)( 21,111)( 22,116)( 23,115)
( 24,117)( 25,113)( 26,112)( 27,114)( 28,119)( 29,118)( 30,120)( 31,125)
( 32,124)( 33,126)( 34,122)( 35,121)( 36,123)( 37, 92)( 38, 91)( 39, 93)
( 40, 98)( 41, 97)( 42, 99)( 43, 95)( 44, 94)( 45, 96)( 46,101)( 47,100)
( 48,102)( 49,107)( 50,106)( 51,108)( 52,104)( 53,103)( 54,105)( 55, 74)
( 56, 73)( 57, 75)( 58, 80)( 59, 79)( 60, 81)( 61, 77)( 62, 76)( 63, 78)
( 64, 83)( 65, 82)( 66, 84)( 67, 89)( 68, 88)( 69, 90)( 70, 86)( 71, 85)
( 72, 87)(145,272)(146,271)(147,273)(148,278)(149,277)(150,279)(151,275)
(152,274)(153,276)(154,281)(155,280)(156,282)(157,287)(158,286)(159,288)
(160,284)(161,283)(162,285)(163,254)(164,253)(165,255)(166,260)(167,259)
(168,261)(169,257)(170,256)(171,258)(172,263)(173,262)(174,264)(175,269)
(176,268)(177,270)(178,266)(179,265)(180,267)(181,236)(182,235)(183,237)
(184,242)(185,241)(186,243)(187,239)(188,238)(189,240)(190,245)(191,244)
(192,246)(193,251)(194,250)(195,252)(196,248)(197,247)(198,249)(199,218)
(200,217)(201,219)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)
(208,227)(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)
(216,231);
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*s2*s1*s2*s0*s1*s2*s0*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, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope