Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,6,6}

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

Overview

Group
SmallGroup(1728,15977)
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

8-fold

9-fold

12-fold

18-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=<(s1*s2)^2> of order 3

10 facets

8 vertex figures

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

6 facets

8 vertex figures

Representations

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

References

None.

to this polytope.