Polytope of Type {4,12,6} with group SmallGroup(1728,47870)

Overview

Group: SmallGroup(1728,47870)
Rank: 4
Schläfli Type: {4,12,6}
Vertices, edges, …: 4, 72, 108, 18
Order of s₀s₁s₂s₃: 12
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

2-fold

{4,12,6}*864d

9-fold

{4,12,2}*192b

18-fold

{4,6,2}*96c

36-fold

{4,3,2}*48

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> of order 3

6 facets

6 of {4,12}*96b

4 vertex figures

4 of 3-fold non-regular quotient of {12,6}*432i

P/N, where N=<(s2*s3)^2> of order 3

6 facets

6 of {4,12}*96b

4 vertex figures

4 of 3-fold non-regular quotient of {12,6}*432i

Representations

Permutation Representation (GAP)

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