Polytope of Type {12,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,8}*576b
if this polytope has a name.
Group : SmallGroup(576,5410)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 36, 144, 24
Order of s0s1s2 : 8
Order of s0s1s2s1 : 12
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 : {12,4}*288
4-fold quotients : {6,4}*144
8-fold quotients : {6,4}*72
9-fold quotients : {4,8}*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 : {12,8}*1152a, {24,8}*1152a, {24,8}*1152d
3-fold covers : {12,8}*1728d, {12,24}*1728m, {12,24}*1728n, {12,8}*1728h, {12,24}*1728t, {12,24}*1728x
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, 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)
(145,253)(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)(181,217)(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 := ( 1,185)( 2,182)( 3,188)( 4,184)( 5,181)( 6,187)( 7,186)( 8,183)
( 9,189)( 10,194)( 11,191)( 12,197)( 13,193)( 14,190)( 15,196)( 16,195)
( 17,192)( 18,198)( 19,203)( 20,200)( 21,206)( 22,202)( 23,199)( 24,205)
( 25,204)( 26,201)( 27,207)( 28,212)( 29,209)( 30,215)( 31,211)( 32,208)
( 33,214)( 34,213)( 35,210)( 36,216)( 37,149)( 38,146)( 39,152)( 40,148)
( 41,145)( 42,151)( 43,150)( 44,147)( 45,153)( 46,158)( 47,155)( 48,161)
( 49,157)( 50,154)( 51,160)( 52,159)( 53,156)( 54,162)( 55,167)( 56,164)
( 57,170)( 58,166)( 59,163)( 60,169)( 61,168)( 62,165)( 63,171)( 64,176)
( 65,173)( 66,179)( 67,175)( 68,172)( 69,178)( 70,177)( 71,174)( 72,180)
( 73,275)( 74,272)( 75,278)( 76,274)( 77,271)( 78,277)( 79,276)( 80,273)
( 81,279)( 82,284)( 83,281)( 84,287)( 85,283)( 86,280)( 87,286)( 88,285)
( 89,282)( 90,288)( 91,257)( 92,254)( 93,260)( 94,256)( 95,253)( 96,259)
( 97,258)( 98,255)( 99,261)(100,266)(101,263)(102,269)(103,265)(104,262)
(105,268)(106,267)(107,264)(108,270)(109,239)(110,236)(111,242)(112,238)
(113,235)(114,241)(115,240)(116,237)(117,243)(118,248)(119,245)(120,251)
(121,247)(122,244)(123,250)(124,249)(125,246)(126,252)(127,221)(128,218)
(129,224)(130,220)(131,217)(132,223)(133,222)(134,219)(135,225)(136,230)
(137,227)(138,233)(139,229)(140,226)(141,232)(142,231)(143,228)(144,234);;
s2 := ( 1, 73)( 2, 74)( 3, 75)( 4, 79)( 5, 80)( 6, 81)( 7, 76)( 8, 77)
( 9, 78)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)( 16, 85)
( 17, 86)( 18, 87)( 19,100)( 20,101)( 21,102)( 22,106)( 23,107)( 24,108)
( 25,103)( 26,104)( 27,105)( 28, 91)( 29, 92)( 30, 93)( 31, 97)( 32, 98)
( 33, 99)( 34, 94)( 35, 95)( 36, 96)( 37,109)( 38,110)( 39,111)( 40,115)
( 41,116)( 42,117)( 43,112)( 44,113)( 45,114)( 46,118)( 47,119)( 48,120)
( 49,124)( 50,125)( 51,126)( 52,121)( 53,122)( 54,123)( 55,136)( 56,137)
( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)( 64,127)
( 65,128)( 66,129)( 67,133)( 68,134)( 69,135)( 70,130)( 71,131)( 72,132)
(145,262)(146,263)(147,264)(148,268)(149,269)(150,270)(151,265)(152,266)
(153,267)(154,253)(155,254)(156,255)(157,259)(158,260)(159,261)(160,256)
(161,257)(162,258)(163,271)(164,272)(165,273)(166,277)(167,278)(168,279)
(169,274)(170,275)(171,276)(172,280)(173,281)(174,282)(175,286)(176,287)
(177,288)(178,283)(179,284)(180,285)(181,217)(182,218)(183,219)(184,223)
(185,224)(186,225)(187,220)(188,221)(189,222)(190,226)(191,227)(192,228)
(193,232)(194,233)(195,234)(196,229)(197,230)(198,231)(199,244)(200,245)
(201,246)(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)(208,235)
(209,236)(210,237)(211,241)(212,242)(213,243)(214,238)(215,239)(216,240);;
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, 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,
s2*s0*s1*s2*s1*s0*s2*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, 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)(145,253)(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)(181,217)(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)!( 1,185)( 2,182)( 3,188)( 4,184)( 5,181)( 6,187)( 7,186)
( 8,183)( 9,189)( 10,194)( 11,191)( 12,197)( 13,193)( 14,190)( 15,196)
( 16,195)( 17,192)( 18,198)( 19,203)( 20,200)( 21,206)( 22,202)( 23,199)
( 24,205)( 25,204)( 26,201)( 27,207)( 28,212)( 29,209)( 30,215)( 31,211)
( 32,208)( 33,214)( 34,213)( 35,210)( 36,216)( 37,149)( 38,146)( 39,152)
( 40,148)( 41,145)( 42,151)( 43,150)( 44,147)( 45,153)( 46,158)( 47,155)
( 48,161)( 49,157)( 50,154)( 51,160)( 52,159)( 53,156)( 54,162)( 55,167)
( 56,164)( 57,170)( 58,166)( 59,163)( 60,169)( 61,168)( 62,165)( 63,171)
( 64,176)( 65,173)( 66,179)( 67,175)( 68,172)( 69,178)( 70,177)( 71,174)
( 72,180)( 73,275)( 74,272)( 75,278)( 76,274)( 77,271)( 78,277)( 79,276)
( 80,273)( 81,279)( 82,284)( 83,281)( 84,287)( 85,283)( 86,280)( 87,286)
( 88,285)( 89,282)( 90,288)( 91,257)( 92,254)( 93,260)( 94,256)( 95,253)
( 96,259)( 97,258)( 98,255)( 99,261)(100,266)(101,263)(102,269)(103,265)
(104,262)(105,268)(106,267)(107,264)(108,270)(109,239)(110,236)(111,242)
(112,238)(113,235)(114,241)(115,240)(116,237)(117,243)(118,248)(119,245)
(120,251)(121,247)(122,244)(123,250)(124,249)(125,246)(126,252)(127,221)
(128,218)(129,224)(130,220)(131,217)(132,223)(133,222)(134,219)(135,225)
(136,230)(137,227)(138,233)(139,229)(140,226)(141,232)(142,231)(143,228)
(144,234);
s2 := Sym(288)!( 1, 73)( 2, 74)( 3, 75)( 4, 79)( 5, 80)( 6, 81)( 7, 76)
( 8, 77)( 9, 78)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)
( 16, 85)( 17, 86)( 18, 87)( 19,100)( 20,101)( 21,102)( 22,106)( 23,107)
( 24,108)( 25,103)( 26,104)( 27,105)( 28, 91)( 29, 92)( 30, 93)( 31, 97)
( 32, 98)( 33, 99)( 34, 94)( 35, 95)( 36, 96)( 37,109)( 38,110)( 39,111)
( 40,115)( 41,116)( 42,117)( 43,112)( 44,113)( 45,114)( 46,118)( 47,119)
( 48,120)( 49,124)( 50,125)( 51,126)( 52,121)( 53,122)( 54,123)( 55,136)
( 56,137)( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)
( 64,127)( 65,128)( 66,129)( 67,133)( 68,134)( 69,135)( 70,130)( 71,131)
( 72,132)(145,262)(146,263)(147,264)(148,268)(149,269)(150,270)(151,265)
(152,266)(153,267)(154,253)(155,254)(156,255)(157,259)(158,260)(159,261)
(160,256)(161,257)(162,258)(163,271)(164,272)(165,273)(166,277)(167,278)
(168,279)(169,274)(170,275)(171,276)(172,280)(173,281)(174,282)(175,286)
(176,287)(177,288)(178,283)(179,284)(180,285)(181,217)(182,218)(183,219)
(184,223)(185,224)(186,225)(187,220)(188,221)(189,222)(190,226)(191,227)
(192,228)(193,232)(194,233)(195,234)(196,229)(197,230)(198,231)(199,244)
(200,245)(201,246)(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)
(208,235)(209,236)(210,237)(211,241)(212,242)(213,243)(214,238)(215,239)
(216,240);
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,
s2*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope