Polytope of Type {6,6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6,4}*1728i
if this polytope has a name.
Group : SmallGroup(1728,47410)
Rank : 5
Schlafli Type : {6,6,6,4}
Number of vertices, edges, etc : 6, 18, 18, 12, 4
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,6,4}*864d, {6,6,6,2}*864g
3-fold quotients : {6,2,6,4}*576a, {6,6,2,4}*576c
4-fold quotients : {3,6,6,2}*432b
6-fold quotients : {3,2,6,4}*288a, {3,6,2,4}*288, {6,2,6,2}*288, {6,6,2,2}*288c
9-fold quotients : {2,2,6,4}*192a, {6,2,2,4}*192
12-fold quotients : {3,2,6,2}*144, {3,6,2,2}*144, {6,2,3,2}*144
18-fold quotients : {3,2,2,4}*96, {2,2,6,2}*96, {6,2,2,2}*96
24-fold quotients : {3,2,3,2}*72
27-fold quotients : {2,2,2,4}*64
36-fold quotients : {2,2,3,2}*48, {3,2,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
None.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213);;
s1 := ( 1,121)( 2,122)( 3,123)( 4,118)( 5,119)( 6,120)( 7,124)( 8,125)( 9,126)( 10,112)( 11,113)( 12,114)( 13,109)( 14,110)( 15,111)( 16,115)( 17,116)( 18,117)( 19,130)( 20,131)( 21,132)( 22,127)( 23,128)( 24,129)( 25,133)( 26,134)( 27,135)( 28,148)( 29,149)( 30,150)( 31,145)( 32,146)( 33,147)( 34,151)( 35,152)( 36,153)( 37,139)( 38,140)( 39,141)( 40,136)( 41,137)( 42,138)( 43,142)( 44,143)( 45,144)( 46,157)( 47,158)( 48,159)( 49,154)( 50,155)( 51,156)( 52,160)( 53,161)( 54,162)( 55,175)( 56,176)( 57,177)( 58,172)( 59,173)( 60,174)( 61,178)( 62,179)( 63,180)( 64,166)( 65,167)( 66,168)( 67,163)( 68,164)( 69,165)( 70,169)( 71,170)( 72,171)( 73,184)( 74,185)( 75,186)( 76,181)( 77,182)( 78,183)( 79,187)( 80,188)( 81,189)( 82,202)( 83,203)( 84,204)( 85,199)( 86,200)( 87,201)( 88,205)( 89,206)( 90,207)( 91,193)( 92,194)( 93,195)( 94,190)( 95,191)( 96,192)( 97,196)( 98,197)( 99,198)(100,211)(101,212)(102,213)(103,208)(104,209)(105,210)(106,214)(107,215)(108,216);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);;
s3 := ( 1, 2)( 4, 5)( 7, 8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)( 46, 47)( 49, 50)( 52, 53)( 55, 83)( 56, 82)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 95)( 68, 94)( 69, 96)( 70, 98)( 71, 97)( 72, 99)( 73,101)( 74,100)( 75,102)( 76,104)( 77,103)( 78,105)( 79,107)( 80,106)( 81,108)(109,110)(112,113)(115,116)(118,119)(121,122)(124,125)(127,128)(130,131)(133,134)(136,137)(139,140)(142,143)(145,146)(148,149)(151,152)(154,155)(157,158)(160,161)(163,191)(164,190)(165,192)(166,194)(167,193)(168,195)(169,197)(170,196)(171,198)(172,200)(173,199)(174,201)(175,203)(176,202)(177,204)(178,206)(179,205)(180,207)(181,209)(182,208)(183,210)(184,212)(185,211)(186,213)(187,215)(188,214)(189,216);;
s4 := ( 1, 55)( 2, 56)( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 4, 7)( 5, 8)( 6, 9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213);
s1 := Sym(216)!( 1,121)( 2,122)( 3,123)( 4,118)( 5,119)( 6,120)( 7,124)( 8,125)( 9,126)( 10,112)( 11,113)( 12,114)( 13,109)( 14,110)( 15,111)( 16,115)( 17,116)( 18,117)( 19,130)( 20,131)( 21,132)( 22,127)( 23,128)( 24,129)( 25,133)( 26,134)( 27,135)( 28,148)( 29,149)( 30,150)( 31,145)( 32,146)( 33,147)( 34,151)( 35,152)( 36,153)( 37,139)( 38,140)( 39,141)( 40,136)( 41,137)( 42,138)( 43,142)( 44,143)( 45,144)( 46,157)( 47,158)( 48,159)( 49,154)( 50,155)( 51,156)( 52,160)( 53,161)( 54,162)( 55,175)( 56,176)( 57,177)( 58,172)( 59,173)( 60,174)( 61,178)( 62,179)( 63,180)( 64,166)( 65,167)( 66,168)( 67,163)( 68,164)( 69,165)( 70,169)( 71,170)( 72,171)( 73,184)( 74,185)( 75,186)( 76,181)( 77,182)( 78,183)( 79,187)( 80,188)( 81,189)( 82,202)( 83,203)( 84,204)( 85,199)( 86,200)( 87,201)( 88,205)( 89,206)( 90,207)( 91,193)( 92,194)( 93,195)( 94,190)( 95,191)( 96,192)( 97,196)( 98,197)( 99,198)(100,211)(101,212)(102,213)(103,208)(104,209)(105,210)(106,214)(107,215)(108,216);
s2 := Sym(216)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);
s3 := Sym(216)!( 1, 2)( 4, 5)( 7, 8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 40, 41)( 43, 44)( 46, 47)( 49, 50)( 52, 53)( 55, 83)( 56, 82)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 95)( 68, 94)( 69, 96)( 70, 98)( 71, 97)( 72, 99)( 73,101)( 74,100)( 75,102)( 76,104)( 77,103)( 78,105)( 79,107)( 80,106)( 81,108)(109,110)(112,113)(115,116)(118,119)(121,122)(124,125)(127,128)(130,131)(133,134)(136,137)(139,140)(142,143)(145,146)(148,149)(151,152)(154,155)(157,158)(160,161)(163,191)(164,190)(165,192)(166,194)(167,193)(168,195)(169,197)(170,196)(171,198)(172,200)(173,199)(174,201)(175,203)(176,202)(177,204)(178,206)(179,205)(180,207)(181,209)(182,208)(183,210)(184,212)(185,211)(186,213)(187,215)(188,214)(189,216);
s4 := Sym(216)!( 1, 55)( 2, 56)( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);
poly := sub<Sym(216)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope