Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,39}

Atlas Canonical Name {4,39}*624

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(624,245)
Rank
3
Schläfli Type
{4,39}
Vertices, edges, …
8, 156, 78
Order of s0s1s2
78
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

12-fold

13-fold

26-fold

52-fold

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*s2*s1*s0*s1*s2> of order 2

52 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1,159)(  2,160)(  3,157)(  4,158)(  5,163)(  6,164)(  7,161)(  8,162)(  9,167)( 10,168)( 11,165)( 12,166)( 13,171)( 14,172)( 15,169)( 16,170)( 17,175)( 18,176)( 19,173)( 20,174)( 21,179)( 22,180)( 23,177)( 24,178)( 25,183)( 26,184)( 27,181)( 28,182)( 29,187)( 30,188)( 31,185)( 32,186)( 33,191)( 34,192)( 35,189)( 36,190)( 37,195)( 38,196)( 39,193)( 40,194)( 41,199)( 42,200)( 43,197)( 44,198)( 45,203)( 46,204)( 47,201)( 48,202)( 49,207)( 50,208)( 51,205)( 52,206)( 53,211)( 54,212)( 55,209)( 56,210)( 57,215)( 58,216)( 59,213)( 60,214)( 61,219)( 62,220)( 63,217)( 64,218)( 65,223)( 66,224)( 67,221)( 68,222)( 69,227)( 70,228)( 71,225)( 72,226)( 73,231)( 74,232)( 75,229)( 76,230)( 77,235)( 78,236)( 79,233)( 80,234)( 81,239)( 82,240)( 83,237)( 84,238)( 85,243)( 86,244)( 87,241)( 88,242)( 89,247)( 90,248)( 91,245)( 92,246)( 93,251)( 94,252)( 95,249)( 96,250)( 97,255)( 98,256)( 99,253)(100,254)(101,259)(102,260)(103,257)(104,258)(105,263)(106,264)(107,261)(108,262)(109,267)(110,268)(111,265)(112,266)(113,271)(114,272)(115,269)(116,270)(117,275)(118,276)(119,273)(120,274)(121,279)(122,280)(123,277)(124,278)(125,283)(126,284)(127,281)(128,282)(129,287)(130,288)(131,285)(132,286)(133,291)(134,292)(135,289)(136,290)(137,295)(138,296)(139,293)(140,294)(141,299)(142,300)(143,297)(144,298)(145,303)(146,304)(147,301)(148,302)(149,307)(150,308)(151,305)(152,306)(153,311)(154,312)(155,309)(156,310);;
s1 := (  2,  3)(  5, 49)(  6, 51)(  7, 50)(  8, 52)(  9, 45)( 10, 47)( 11, 46)( 12, 48)( 13, 41)( 14, 43)( 15, 42)( 16, 44)( 17, 37)( 18, 39)( 19, 38)( 20, 40)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 25, 29)( 26, 31)( 27, 30)( 28, 32)( 53,105)( 54,107)( 55,106)( 56,108)( 57,153)( 58,155)( 59,154)( 60,156)( 61,149)( 62,151)( 63,150)( 64,152)( 65,145)( 66,147)( 67,146)( 68,148)( 69,141)( 70,143)( 71,142)( 72,144)( 73,137)( 74,139)( 75,138)( 76,140)( 77,133)( 78,135)( 79,134)( 80,136)( 81,129)( 82,131)( 83,130)( 84,132)( 85,125)( 86,127)( 87,126)( 88,128)( 89,121)( 90,123)( 91,122)( 92,124)( 93,117)( 94,119)( 95,118)( 96,120)( 97,113)( 98,115)( 99,114)(100,116)(101,109)(102,111)(103,110)(104,112)(158,159)(161,205)(162,207)(163,206)(164,208)(165,201)(166,203)(167,202)(168,204)(169,197)(170,199)(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,189)(178,191)(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(209,261)(210,263)(211,262)(212,264)(213,309)(214,311)(215,310)(216,312)(217,305)(218,307)(219,306)(220,308)(221,301)(222,303)(223,302)(224,304)(225,297)(226,299)(227,298)(228,300)(229,293)(230,295)(231,294)(232,296)(233,289)(234,291)(235,290)(236,292)(237,285)(238,287)(239,286)(240,288)(241,281)(242,283)(243,282)(244,284)(245,277)(246,279)(247,278)(248,280)(249,273)(250,275)(251,274)(252,276)(253,269)(254,271)(255,270)(256,272)(257,265)(258,267)(259,266)(260,268);;
s2 := (  1,109)(  2,112)(  3,111)(  4,110)(  5,105)(  6,108)(  7,107)(  8,106)(  9,153)( 10,156)( 11,155)( 12,154)( 13,149)( 14,152)( 15,151)( 16,150)( 17,145)( 18,148)( 19,147)( 20,146)( 21,141)( 22,144)( 23,143)( 24,142)( 25,137)( 26,140)( 27,139)( 28,138)( 29,133)( 30,136)( 31,135)( 32,134)( 33,129)( 34,132)( 35,131)( 36,130)( 37,125)( 38,128)( 39,127)( 40,126)( 41,121)( 42,124)( 43,123)( 44,122)( 45,117)( 46,120)( 47,119)( 48,118)( 49,113)( 50,116)( 51,115)( 52,114)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 61,101)( 62,104)( 63,103)( 64,102)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)( 71, 95)( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)( 79, 87)( 80, 86)( 82, 84)(157,265)(158,268)(159,267)(160,266)(161,261)(162,264)(163,263)(164,262)(165,309)(166,312)(167,311)(168,310)(169,305)(170,308)(171,307)(172,306)(173,301)(174,304)(175,303)(176,302)(177,297)(178,300)(179,299)(180,298)(181,293)(182,296)(183,295)(184,294)(185,289)(186,292)(187,291)(188,290)(189,285)(190,288)(191,287)(192,286)(193,281)(194,284)(195,283)(196,282)(197,277)(198,280)(199,279)(200,278)(201,273)(202,276)(203,275)(204,274)(205,269)(206,272)(207,271)(208,270)(209,213)(210,216)(211,215)(212,214)(217,257)(218,260)(219,259)(220,258)(221,253)(222,256)(223,255)(224,254)(225,249)(226,252)(227,251)(228,250)(229,245)(230,248)(231,247)(232,246)(233,241)(234,244)(235,243)(236,242)(238,240);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(312)!(  1,159)(  2,160)(  3,157)(  4,158)(  5,163)(  6,164)(  7,161)(  8,162)(  9,167)( 10,168)( 11,165)( 12,166)( 13,171)( 14,172)( 15,169)( 16,170)( 17,175)( 18,176)( 19,173)( 20,174)( 21,179)( 22,180)( 23,177)( 24,178)( 25,183)( 26,184)( 27,181)( 28,182)( 29,187)( 30,188)( 31,185)( 32,186)( 33,191)( 34,192)( 35,189)( 36,190)( 37,195)( 38,196)( 39,193)( 40,194)( 41,199)( 42,200)( 43,197)( 44,198)( 45,203)( 46,204)( 47,201)( 48,202)( 49,207)( 50,208)( 51,205)( 52,206)( 53,211)( 54,212)( 55,209)( 56,210)( 57,215)( 58,216)( 59,213)( 60,214)( 61,219)( 62,220)( 63,217)( 64,218)( 65,223)( 66,224)( 67,221)( 68,222)( 69,227)( 70,228)( 71,225)( 72,226)( 73,231)( 74,232)( 75,229)( 76,230)( 77,235)( 78,236)( 79,233)( 80,234)( 81,239)( 82,240)( 83,237)( 84,238)( 85,243)( 86,244)( 87,241)( 88,242)( 89,247)( 90,248)( 91,245)( 92,246)( 93,251)( 94,252)( 95,249)( 96,250)( 97,255)( 98,256)( 99,253)(100,254)(101,259)(102,260)(103,257)(104,258)(105,263)(106,264)(107,261)(108,262)(109,267)(110,268)(111,265)(112,266)(113,271)(114,272)(115,269)(116,270)(117,275)(118,276)(119,273)(120,274)(121,279)(122,280)(123,277)(124,278)(125,283)(126,284)(127,281)(128,282)(129,287)(130,288)(131,285)(132,286)(133,291)(134,292)(135,289)(136,290)(137,295)(138,296)(139,293)(140,294)(141,299)(142,300)(143,297)(144,298)(145,303)(146,304)(147,301)(148,302)(149,307)(150,308)(151,305)(152,306)(153,311)(154,312)(155,309)(156,310);
s1 := Sym(312)!(  2,  3)(  5, 49)(  6, 51)(  7, 50)(  8, 52)(  9, 45)( 10, 47)( 11, 46)( 12, 48)( 13, 41)( 14, 43)( 15, 42)( 16, 44)( 17, 37)( 18, 39)( 19, 38)( 20, 40)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 25, 29)( 26, 31)( 27, 30)( 28, 32)( 53,105)( 54,107)( 55,106)( 56,108)( 57,153)( 58,155)( 59,154)( 60,156)( 61,149)( 62,151)( 63,150)( 64,152)( 65,145)( 66,147)( 67,146)( 68,148)( 69,141)( 70,143)( 71,142)( 72,144)( 73,137)( 74,139)( 75,138)( 76,140)( 77,133)( 78,135)( 79,134)( 80,136)( 81,129)( 82,131)( 83,130)( 84,132)( 85,125)( 86,127)( 87,126)( 88,128)( 89,121)( 90,123)( 91,122)( 92,124)( 93,117)( 94,119)( 95,118)( 96,120)( 97,113)( 98,115)( 99,114)(100,116)(101,109)(102,111)(103,110)(104,112)(158,159)(161,205)(162,207)(163,206)(164,208)(165,201)(166,203)(167,202)(168,204)(169,197)(170,199)(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,189)(178,191)(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(209,261)(210,263)(211,262)(212,264)(213,309)(214,311)(215,310)(216,312)(217,305)(218,307)(219,306)(220,308)(221,301)(222,303)(223,302)(224,304)(225,297)(226,299)(227,298)(228,300)(229,293)(230,295)(231,294)(232,296)(233,289)(234,291)(235,290)(236,292)(237,285)(238,287)(239,286)(240,288)(241,281)(242,283)(243,282)(244,284)(245,277)(246,279)(247,278)(248,280)(249,273)(250,275)(251,274)(252,276)(253,269)(254,271)(255,270)(256,272)(257,265)(258,267)(259,266)(260,268);
s2 := Sym(312)!(  1,109)(  2,112)(  3,111)(  4,110)(  5,105)(  6,108)(  7,107)(  8,106)(  9,153)( 10,156)( 11,155)( 12,154)( 13,149)( 14,152)( 15,151)( 16,150)( 17,145)( 18,148)( 19,147)( 20,146)( 21,141)( 22,144)( 23,143)( 24,142)( 25,137)( 26,140)( 27,139)( 28,138)( 29,133)( 30,136)( 31,135)( 32,134)( 33,129)( 34,132)( 35,131)( 36,130)( 37,125)( 38,128)( 39,127)( 40,126)( 41,121)( 42,124)( 43,123)( 44,122)( 45,117)( 46,120)( 47,119)( 48,118)( 49,113)( 50,116)( 51,115)( 52,114)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 61,101)( 62,104)( 63,103)( 64,102)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)( 71, 95)( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)( 79, 87)( 80, 86)( 82, 84)(157,265)(158,268)(159,267)(160,266)(161,261)(162,264)(163,263)(164,262)(165,309)(166,312)(167,311)(168,310)(169,305)(170,308)(171,307)(172,306)(173,301)(174,304)(175,303)(176,302)(177,297)(178,300)(179,299)(180,298)(181,293)(182,296)(183,295)(184,294)(185,289)(186,292)(187,291)(188,290)(189,285)(190,288)(191,287)(192,286)(193,281)(194,284)(195,283)(196,282)(197,277)(198,280)(199,279)(200,278)(201,273)(202,276)(203,275)(204,274)(205,269)(206,272)(207,271)(208,270)(209,213)(210,216)(211,215)(212,214)(217,257)(218,260)(219,259)(220,258)(221,253)(222,256)(223,255)(224,254)(225,249)(226,252)(227,251)(228,250)(229,245)(230,248)(231,247)(232,246)(233,241)(234,244)(235,243)(236,242)(238,240);
poly := sub<Sym(312)|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*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.

Twisty Puzzle