Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,6,6}

Atlas Canonical Name {8,6,6}*1728e

Overview

Group
SmallGroup(1728,37593)
Rank
4
Schläfli Type
{8,6,6}
Vertices, edges, …
8, 72, 54, 18
Order of s0s1s2s3
24
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

24-fold

27-fold

36-fold

54-fold

72-fold

108-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s2*s1*(s2*s3)^2*s2*s1*s3*s2> of order 2

9 facets

8 vertex figures

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

12 facets

8 vertex figures

P/N, where N=<s1*s2*s3*s2*s1*s3> of order 3

6 facets

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189);;
s1 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)( 70,103)( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)(109,163)(110,164)(111,165)(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,181)(119,182)(120,183)(121,187)(122,188)(123,189)(124,184)(125,185)(126,186)(127,172)(128,173)(129,174)(130,178)(131,179)(132,180)(133,175)(134,176)(135,177)(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)(144,195)(145,208)(146,209)(147,210)(148,214)(149,215)(150,216)(151,211)(152,212)(153,213)(154,199)(155,200)(156,201)(157,205)(158,206)(159,207)(160,202)(161,203)(162,204);;
s2 := (  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)(  8, 18)(  9, 17)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 49)( 47, 51)( 48, 50)( 53, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)(100,103)(101,105)(102,104)(107,108)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,130)(128,132)(129,131)(134,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)(143,153)(144,152)(154,157)(155,159)(156,158)(161,162)(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,184)(182,186)(183,185)(188,189)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(196,205)(197,207)(198,206)(208,211)(209,213)(210,212)(215,216);;
s3 := (  1,  2)(  4,  5)(  7,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 28, 29)( 31, 32)( 34, 35)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 55, 56)( 58, 59)( 61, 62)( 64, 74)( 65, 73)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 82, 83)( 85, 86)( 88, 89)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(109,110)(112,113)(115,116)(118,128)(119,127)(120,129)(121,131)(122,130)(123,132)(124,134)(125,133)(126,135)(136,137)(139,140)(142,143)(145,155)(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(163,164)(166,167)(169,170)(172,182)(173,181)(174,183)(175,185)(176,184)(177,186)(178,188)(179,187)(180,189)(190,191)(193,194)(196,197)(199,209)(200,208)(201,210)(202,212)(203,211)(204,213)(205,215)(206,214)(207,216);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(216)!(  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189);
s1 := Sym(216)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)( 70,103)( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)(109,163)(110,164)(111,165)(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,181)(119,182)(120,183)(121,187)(122,188)(123,189)(124,184)(125,185)(126,186)(127,172)(128,173)(129,174)(130,178)(131,179)(132,180)(133,175)(134,176)(135,177)(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)(144,195)(145,208)(146,209)(147,210)(148,214)(149,215)(150,216)(151,211)(152,212)(153,213)(154,199)(155,200)(156,201)(157,205)(158,206)(159,207)(160,202)(161,203)(162,204);
s2 := Sym(216)!(  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 16)(  8, 18)(  9, 17)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 49)( 47, 51)( 48, 50)( 53, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)(100,103)(101,105)(102,104)(107,108)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,130)(128,132)(129,131)(134,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)(143,153)(144,152)(154,157)(155,159)(156,158)(161,162)(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,184)(182,186)(183,185)(188,189)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(196,205)(197,207)(198,206)(208,211)(209,213)(210,212)(215,216);
s3 := Sym(216)!(  1,  2)(  4,  5)(  7,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 28, 29)( 31, 32)( 34, 35)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 55, 56)( 58, 59)( 61, 62)( 64, 74)( 65, 73)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 82, 83)( 85, 86)( 88, 89)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(109,110)(112,113)(115,116)(118,128)(119,127)(120,129)(121,131)(122,130)(123,132)(124,134)(125,133)(126,135)(136,137)(139,140)(142,143)(145,155)(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(163,164)(166,167)(169,170)(172,182)(173,181)(174,183)(175,185)(176,184)(177,186)(178,188)(179,187)(180,189)(190,191)(193,194)(196,197)(199,209)(200,208)(201,210)(202,212)(203,211)(204,213)(205,215)(206,214)(207,216);
poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.