Polytope of Type {6,6,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4,3}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {6,6,4,3}
Number of vertices, edges, etc : 6, 18, 24, 12, 6
Order of s0s1s2s3s4 : 6
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 : {3,6,4,3}*864
3-fold quotients : {6,2,4,3}*576
4-fold quotients : {6,6,2,3}*432c
6-fold quotients : {3,2,4,3}*288, {6,2,4,3}*288
8-fold quotients : {3,6,2,3}*216
9-fold quotients : {2,2,4,3}*192
12-fold quotients : {3,2,4,3}*144, {6,2,2,3}*144
18-fold quotients : {2,2,4,3}*96
24-fold quotients : {3,2,2,3}*72
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s2*s3*s2*s3> of order 2.
4 facets:
2 of {6,6,2}*144c
2 of {6,6,4}*288c
6 vertex figures:
6 of 2-fold non-regular quotient of {6,4,3}*288
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 := ( 1,125)( 2,126)( 3,127)( 4,128)( 5,121)( 6,122)( 7,123)( 8,124)( 9,129)( 10,130)( 11,131)( 12,132)( 13,113)( 14,114)( 15,115)( 16,116)( 17,109)( 18,110)( 19,111)( 20,112)( 21,117)( 22,118)( 23,119)( 24,120)( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)( 32,136)( 33,141)( 34,142)( 35,143)( 36,144)( 37,161)( 38,162)( 39,163)( 40,164)( 41,157)( 42,158)( 43,159)( 44,160)( 45,165)( 46,166)( 47,167)( 48,168)( 49,149)( 50,150)( 51,151)( 52,152)( 53,145)( 54,146)( 55,147)( 56,148)( 57,153)( 58,154)( 59,155)( 60,156)( 61,173)( 62,174)( 63,175)( 64,176)( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)( 72,180)( 73,197)( 74,198)( 75,199)( 76,200)( 77,193)( 78,194)( 79,195)( 80,196)( 81,201)( 82,202)( 83,203)( 84,204)( 85,185)( 86,186)( 87,187)( 88,188)( 89,181)( 90,182)( 91,183)( 92,184)( 93,189)( 94,190)( 95,191)( 96,192)( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)(104,208)(105,213)(106,214)(107,215)(108,216);;
s2 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)(158,172)(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)(166,180)(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)(200,210)(201,215)(202,216)(203,213)(204,214);;
s3 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 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, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)(179,216)(180,215);;
s4 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)( 8, 42)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)( 16, 50)( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)( 32, 66)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)( 86, 88)( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)(127,163)(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)(135,171)(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)(143,179)(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(206,208)(210,212)(214,216);;
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, s3*s4*s3*s4*s3*s4,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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 ];;
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)!( 1,125)( 2,126)( 3,127)( 4,128)( 5,121)( 6,122)( 7,123)( 8,124)( 9,129)( 10,130)( 11,131)( 12,132)( 13,113)( 14,114)( 15,115)( 16,116)( 17,109)( 18,110)( 19,111)( 20,112)( 21,117)( 22,118)( 23,119)( 24,120)( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)( 32,136)( 33,141)( 34,142)( 35,143)( 36,144)( 37,161)( 38,162)( 39,163)( 40,164)( 41,157)( 42,158)( 43,159)( 44,160)( 45,165)( 46,166)( 47,167)( 48,168)( 49,149)( 50,150)( 51,151)( 52,152)( 53,145)( 54,146)( 55,147)( 56,148)( 57,153)( 58,154)( 59,155)( 60,156)( 61,173)( 62,174)( 63,175)( 64,176)( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)( 72,180)( 73,197)( 74,198)( 75,199)( 76,200)( 77,193)( 78,194)( 79,195)( 80,196)( 81,201)( 82,202)( 83,203)( 84,204)( 85,185)( 86,186)( 87,187)( 88,188)( 89,181)( 90,182)( 91,183)( 92,184)( 93,189)( 94,190)( 95,191)( 96,192)( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)(104,208)(105,213)(106,214)(107,215)(108,216);
s2 := Sym(216)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)(158,172)(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)(166,180)(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)(200,210)(201,215)(202,216)(203,213)(204,214);
s3 := Sym(216)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 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, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)(179,216)(180,215);
s4 := Sym(216)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)( 8, 42)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)( 16, 50)( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)( 32, 66)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)( 86, 88)( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)(127,163)(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)(135,171)(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)(143,179)(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(206,208)(210,212)(214,216);
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, s3*s4*s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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 >;
References : None.
to this polytope