Polytope of Type {4,6,6} with group SmallGroup(864,4000)

Overview

Group: SmallGroup(864,4000)
Rank: 4
Schläfli Type: {4,6,6}
Vertices, edges, …: 4, 36, 54, 18
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Non-Orientable
Flat

Quotients maximal quotients in bold

2-fold

{4,3,6}*432

3-fold

{4,6,6}*288f

6-fold

{4,3,6}*144

9-fold

{4,6,2}*96b

18-fold

{4,3,2}*48

Covers minimal covers in bold

2-fold

{4,6,6}*1728a

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*s3*s2)^2> of order 3

6 facets

6 of {4,6}*48b

4 vertex figures

4 of 3-fold non-regular quotient of {6,6}*216c

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.