Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6,6}

Atlas Canonical Name {6,6,6}*1728f

Overview

Group
SmallGroup(1728,47874)
Rank
4
Schläfli Type
{6,6,6}
Vertices, edges, …
24, 72, 72, 6
Order of s0s1s2s3
12
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

24-fold

36-fold

48-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=<(s0*s1)^2*(s2*s1)^2*s0*s2*s1*s2> of order 2

6 facets

12 vertex figures

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

6 facets

12 vertex figures

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

6 facets

12 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*(s1*s2)^3> of order 4

6 facets

6 vertex figures

Representations

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

References

None.

to this polytope.