Polytope of Type {6,12,6} with group SmallGroup(1728,47874)

Overview

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

{6,12,6}*864h

3-fold

{2,12,6}*576a

4-fold

{6,6,6}*432d

6-fold

{2,12,6}*288d

9-fold

{2,4,6}*192

12-fold

18-fold

24-fold

{6,3,2}*72

36-fold

72-fold

108-fold

{2,2,2}*16

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*(s3*s2)^2*s1*(s2*s3)^2> of order 2

8 facets

4 of {6,6}*72b
4 of {6,12}*144b

6 vertex figures

6 of 2-fold non-regular quotient of {12,6}*288a

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.