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

Overview

Group: SmallGroup(864,4000)
Rank: 4
Schläfli Type: {6,6,4}
Vertices, edges, …: 18, 54, 36, 4
Order of s₀s₁s₂s₃: 6
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

{6,6,4}*432

3-fold

{6,6,4}*288d

9-fold

{2,6,4}*96c

18-fold

{2,3,4}*48

Covers minimal covers in bold

2-fold

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> of order 3

4 facets

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

6 vertex figures

6 of {6,4}*48c

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.