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# Polytope of Type {12,8}

Atlas Canonical Name : {12,8}*576a
if this polytope has a name.
Group : SmallGroup(576,5307)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 36, 144, 24
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,8,2} of size 1152
Vertex Figure Of :
{2,12,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,8}*288, {12,4}*288
4-fold quotients : {6,4}*144
8-fold quotients : {6,4}*72
9-fold quotients : {4,8}*64a
18-fold quotients : {4,4}*32, {2,8}*32
36-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,8}*1152a, {24,8}*1152b, {24,8}*1152c, {12,16}*1152a, {12,16}*1152b
3-fold covers : {12,8}*1728a, {12,24}*1728g, {12,24}*1728h, {12,8}*1728g, {12,24}*1728s, {12,24}*1728u
Permutation Representation (GAP) :
s0 := (  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)
(  9,113)( 10,118)( 11,120)( 12,119)( 13,124)( 14,126)( 15,125)( 16,121)
( 17,123)( 18,122)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)( 24,134)
( 25,130)( 26,132)( 27,131)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)
( 33,143)( 34,139)( 35,141)( 36,140)( 37, 82)( 38, 84)( 39, 83)( 40, 88)
( 41, 90)( 42, 89)( 43, 85)( 44, 87)( 45, 86)( 46, 73)( 47, 75)( 48, 74)
( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 55,100)( 56,102)
( 57,101)( 58,106)( 59,108)( 60,107)( 61,103)( 62,105)( 63,104)( 64, 91)
( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)( 71, 96)( 72, 95)
(145,253)(146,255)(147,254)(148,259)(149,261)(150,260)(151,256)(152,258)
(153,257)(154,262)(155,264)(156,263)(157,268)(158,270)(159,269)(160,265)
(161,267)(162,266)(163,271)(164,273)(165,272)(166,277)(167,279)(168,278)
(169,274)(170,276)(171,275)(172,280)(173,282)(174,281)(175,286)(176,288)
(177,287)(178,283)(179,285)(180,284)(181,226)(182,228)(183,227)(184,232)
(185,234)(186,233)(187,229)(188,231)(189,230)(190,217)(191,219)(192,218)
(193,223)(194,225)(195,224)(196,220)(197,222)(198,221)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,235)
(209,237)(210,236)(211,241)(212,243)(213,242)(214,238)(215,240)(216,239);;
s1 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 49)( 38, 50)( 39, 51)( 40, 46)
( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)( 55, 67)( 56, 68)( 57, 69)
( 58, 64)( 59, 65)( 60, 66)( 61, 70)( 62, 71)( 63, 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)
(109,139)(110,140)(111,141)(112,136)(113,137)(114,138)(115,142)(116,143)
(117,144)(118,130)(119,131)(120,132)(121,127)(122,128)(123,129)(124,133)
(125,134)(126,135)(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)(163,202)(164,203)(165,204)(166,199)
(167,200)(168,201)(169,205)(170,206)(171,207)(172,211)(173,212)(174,213)
(175,208)(176,209)(177,210)(178,214)(179,215)(180,216)(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)( 19,163)( 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)
( 73,226)( 74,229)( 75,232)( 76,227)( 77,230)( 78,233)( 79,228)( 80,231)
( 81,234)( 82,217)( 83,220)( 84,223)( 85,218)( 86,221)( 87,224)( 88,219)
( 89,222)( 90,225)( 91,244)( 92,247)( 93,250)( 94,245)( 95,248)( 96,251)
( 97,246)( 98,249)( 99,252)(100,235)(101,238)(102,241)(103,236)(104,239)
(105,242)(106,237)(107,240)(108,243)(109,253)(110,256)(111,259)(112,254)
(113,257)(114,260)(115,255)(116,258)(117,261)(118,262)(119,265)(120,268)
(121,263)(122,266)(123,269)(124,264)(125,267)(126,270)(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, 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,
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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope