Polytope of Type {12,8} with group SmallGroup(576,5410)

Overview

Group: SmallGroup(576,5410)
Rank: 3
Schläfli Type: {12,8}
Vertices, edges, …: 36, 144, 24
Order of s₀s₁s₂: 8
Order of s₀s₁s₂s₁: 12
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{12,4}*288

4-fold

{6,4}*144

8-fold

{6,4}*72

9-fold

{4,8}*64b

18-fold

{4,4}*32

36-fold

72-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0*s1> of order 2

12 facets

12 of {12}*24

18 vertex figures

18 of {8}*16

P/N, where N=<(s0*s1)^4> of order 3

16 facets

12 of {4}*8
4 of {12}*24

12 vertex figures

12 of {8}*16

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3

8 facets

8 of {12}*24

12 vertex figures

12 of {8}*16

Representations

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, s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*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, 
s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2 ];;
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, s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*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, 
s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2 >;

References

None.

to this polytope.