Polytope of Type {6,6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6,3}*1728a
Also Known As : {{6,6|2},{6,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {6,6,6,3}
Number of vertices, edges, etc : 6, 18, 24, 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) :
3-fold quotients : {2,6,6,3}*576, {6,2,6,3}*576
6-fold quotients : {3,2,6,3}*288, {6,2,3,3}*288
9-fold quotients : {2,2,6,3}*192
12-fold quotients : {3,2,3,3}*144
18-fold quotients : {2,2,3,3}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216);;
s1 := ( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 21)( 6, 22)( 7, 23)( 8, 24)
( 9, 17)( 10, 18)( 11, 19)( 12, 20)( 29, 33)( 30, 34)( 31, 35)( 32, 36)
( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 57)( 42, 58)( 43, 59)( 44, 60)
( 45, 53)( 46, 54)( 47, 55)( 48, 56)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 73, 85)( 74, 86)( 75, 87)( 76, 88)( 77, 93)( 78, 94)( 79, 95)( 80, 96)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)(101,105)(102,106)(103,107)(104,108)
(109,121)(110,122)(111,123)(112,124)(113,129)(114,130)(115,131)(116,132)
(117,125)(118,126)(119,127)(120,128)(137,141)(138,142)(139,143)(140,144)
(145,157)(146,158)(147,159)(148,160)(149,165)(150,166)(151,167)(152,168)
(153,161)(154,162)(155,163)(156,164)(173,177)(174,178)(175,179)(176,180)
(181,193)(182,194)(183,195)(184,196)(185,201)(186,202)(187,203)(188,204)
(189,197)(190,198)(191,199)(192,200)(209,213)(210,214)(211,215)(212,216);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 11, 12)( 13, 17)( 14, 18)( 15, 20)
( 16, 19)( 23, 24)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 37, 77)
( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 49, 89)( 50, 90)( 51, 92)( 52, 91)( 53, 85)
( 54, 86)( 55, 88)( 56, 87)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61,101)
( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)
( 70,106)( 71,108)( 72,107)(109,113)(110,114)(111,116)(112,115)(119,120)
(121,125)(122,126)(123,128)(124,127)(131,132)(133,137)(134,138)(135,140)
(136,139)(143,144)(145,185)(146,186)(147,188)(148,187)(149,181)(150,182)
(151,184)(152,183)(153,189)(154,190)(155,192)(156,191)(157,197)(158,198)
(159,200)(160,199)(161,193)(162,194)(163,196)(164,195)(165,201)(166,202)
(167,204)(168,203)(169,209)(170,210)(171,212)(172,211)(173,205)(174,206)
(175,208)(176,207)(177,213)(178,214)(179,216)(180,215);;
s3 := ( 1,145)( 2,148)( 3,147)( 4,146)( 5,149)( 6,152)( 7,151)( 8,150)
( 9,153)( 10,156)( 11,155)( 12,154)( 13,157)( 14,160)( 15,159)( 16,158)
( 17,161)( 18,164)( 19,163)( 20,162)( 21,165)( 22,168)( 23,167)( 24,166)
( 25,169)( 26,172)( 27,171)( 28,170)( 29,173)( 30,176)( 31,175)( 32,174)
( 33,177)( 34,180)( 35,179)( 36,178)( 37,109)( 38,112)( 39,111)( 40,110)
( 41,113)( 42,116)( 43,115)( 44,114)( 45,117)( 46,120)( 47,119)( 48,118)
( 49,121)( 50,124)( 51,123)( 52,122)( 53,125)( 54,128)( 55,127)( 56,126)
( 57,129)( 58,132)( 59,131)( 60,130)( 61,133)( 62,136)( 63,135)( 64,134)
( 65,137)( 66,140)( 67,139)( 68,138)( 69,141)( 70,144)( 71,143)( 72,142)
( 73,181)( 74,184)( 75,183)( 76,182)( 77,185)( 78,188)( 79,187)( 80,186)
( 81,189)( 82,192)( 83,191)( 84,190)( 85,193)( 86,196)( 87,195)( 88,194)
( 89,197)( 90,200)( 91,199)( 92,198)( 93,201)( 94,204)( 95,203)( 96,202)
( 97,205)( 98,208)( 99,207)(100,206)(101,209)(102,212)(103,211)(104,210)
(105,213)(106,216)(107,215)(108,214);;
s4 := ( 1,110)( 2,109)( 3,111)( 4,112)( 5,114)( 6,113)( 7,115)( 8,116)
( 9,118)( 10,117)( 11,119)( 12,120)( 13,122)( 14,121)( 15,123)( 16,124)
( 17,126)( 18,125)( 19,127)( 20,128)( 21,130)( 22,129)( 23,131)( 24,132)
( 25,134)( 26,133)( 27,135)( 28,136)( 29,138)( 30,137)( 31,139)( 32,140)
( 33,142)( 34,141)( 35,143)( 36,144)( 37,182)( 38,181)( 39,183)( 40,184)
( 41,186)( 42,185)( 43,187)( 44,188)( 45,190)( 46,189)( 47,191)( 48,192)
( 49,194)( 50,193)( 51,195)( 52,196)( 53,198)( 54,197)( 55,199)( 56,200)
( 57,202)( 58,201)( 59,203)( 60,204)( 61,206)( 62,205)( 63,207)( 64,208)
( 65,210)( 66,209)( 67,211)( 68,212)( 69,214)( 70,213)( 71,215)( 72,216)
( 73,146)( 74,145)( 75,147)( 76,148)( 77,150)( 78,149)( 79,151)( 80,152)
( 81,154)( 82,153)( 83,155)( 84,156)( 85,158)( 86,157)( 87,159)( 88,160)
( 89,162)( 90,161)( 91,163)( 92,164)( 93,166)( 94,165)( 95,167)( 96,168)
( 97,170)( 98,169)( 99,171)(100,172)(101,174)(102,173)(103,175)(104,176)
(105,178)(106,177)(107,179)(108,180);;
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)
( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)
(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)
(132,144)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)
(164,176)(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)
(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)
(204,216);
s1 := Sym(216)!( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 21)( 6, 22)( 7, 23)
( 8, 24)( 9, 17)( 10, 18)( 11, 19)( 12, 20)( 29, 33)( 30, 34)( 31, 35)
( 32, 36)( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 57)( 42, 58)( 43, 59)
( 44, 60)( 45, 53)( 46, 54)( 47, 55)( 48, 56)( 65, 69)( 66, 70)( 67, 71)
( 68, 72)( 73, 85)( 74, 86)( 75, 87)( 76, 88)( 77, 93)( 78, 94)( 79, 95)
( 80, 96)( 81, 89)( 82, 90)( 83, 91)( 84, 92)(101,105)(102,106)(103,107)
(104,108)(109,121)(110,122)(111,123)(112,124)(113,129)(114,130)(115,131)
(116,132)(117,125)(118,126)(119,127)(120,128)(137,141)(138,142)(139,143)
(140,144)(145,157)(146,158)(147,159)(148,160)(149,165)(150,166)(151,167)
(152,168)(153,161)(154,162)(155,163)(156,164)(173,177)(174,178)(175,179)
(176,180)(181,193)(182,194)(183,195)(184,196)(185,201)(186,202)(187,203)
(188,204)(189,197)(190,198)(191,199)(192,200)(209,213)(210,214)(211,215)
(212,216);
s2 := Sym(216)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 11, 12)( 13, 17)( 14, 18)
( 15, 20)( 16, 19)( 23, 24)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)
( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)
( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 89)( 50, 90)( 51, 92)( 52, 91)
( 53, 85)( 54, 86)( 55, 88)( 56, 87)( 57, 93)( 58, 94)( 59, 96)( 60, 95)
( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)
( 69,105)( 70,106)( 71,108)( 72,107)(109,113)(110,114)(111,116)(112,115)
(119,120)(121,125)(122,126)(123,128)(124,127)(131,132)(133,137)(134,138)
(135,140)(136,139)(143,144)(145,185)(146,186)(147,188)(148,187)(149,181)
(150,182)(151,184)(152,183)(153,189)(154,190)(155,192)(156,191)(157,197)
(158,198)(159,200)(160,199)(161,193)(162,194)(163,196)(164,195)(165,201)
(166,202)(167,204)(168,203)(169,209)(170,210)(171,212)(172,211)(173,205)
(174,206)(175,208)(176,207)(177,213)(178,214)(179,216)(180,215);
s3 := Sym(216)!( 1,145)( 2,148)( 3,147)( 4,146)( 5,149)( 6,152)( 7,151)
( 8,150)( 9,153)( 10,156)( 11,155)( 12,154)( 13,157)( 14,160)( 15,159)
( 16,158)( 17,161)( 18,164)( 19,163)( 20,162)( 21,165)( 22,168)( 23,167)
( 24,166)( 25,169)( 26,172)( 27,171)( 28,170)( 29,173)( 30,176)( 31,175)
( 32,174)( 33,177)( 34,180)( 35,179)( 36,178)( 37,109)( 38,112)( 39,111)
( 40,110)( 41,113)( 42,116)( 43,115)( 44,114)( 45,117)( 46,120)( 47,119)
( 48,118)( 49,121)( 50,124)( 51,123)( 52,122)( 53,125)( 54,128)( 55,127)
( 56,126)( 57,129)( 58,132)( 59,131)( 60,130)( 61,133)( 62,136)( 63,135)
( 64,134)( 65,137)( 66,140)( 67,139)( 68,138)( 69,141)( 70,144)( 71,143)
( 72,142)( 73,181)( 74,184)( 75,183)( 76,182)( 77,185)( 78,188)( 79,187)
( 80,186)( 81,189)( 82,192)( 83,191)( 84,190)( 85,193)( 86,196)( 87,195)
( 88,194)( 89,197)( 90,200)( 91,199)( 92,198)( 93,201)( 94,204)( 95,203)
( 96,202)( 97,205)( 98,208)( 99,207)(100,206)(101,209)(102,212)(103,211)
(104,210)(105,213)(106,216)(107,215)(108,214);
s4 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,112)( 5,114)( 6,113)( 7,115)
( 8,116)( 9,118)( 10,117)( 11,119)( 12,120)( 13,122)( 14,121)( 15,123)
( 16,124)( 17,126)( 18,125)( 19,127)( 20,128)( 21,130)( 22,129)( 23,131)
( 24,132)( 25,134)( 26,133)( 27,135)( 28,136)( 29,138)( 30,137)( 31,139)
( 32,140)( 33,142)( 34,141)( 35,143)( 36,144)( 37,182)( 38,181)( 39,183)
( 40,184)( 41,186)( 42,185)( 43,187)( 44,188)( 45,190)( 46,189)( 47,191)
( 48,192)( 49,194)( 50,193)( 51,195)( 52,196)( 53,198)( 54,197)( 55,199)
( 56,200)( 57,202)( 58,201)( 59,203)( 60,204)( 61,206)( 62,205)( 63,207)
( 64,208)( 65,210)( 66,209)( 67,211)( 68,212)( 69,214)( 70,213)( 71,215)
( 72,216)( 73,146)( 74,145)( 75,147)( 76,148)( 77,150)( 78,149)( 79,151)
( 80,152)( 81,154)( 82,153)( 83,155)( 84,156)( 85,158)( 86,157)( 87,159)
( 88,160)( 89,162)( 90,161)( 91,163)( 92,164)( 93,166)( 94,165)( 95,167)
( 96,168)( 97,170)( 98,169)( 99,171)(100,172)(101,174)(102,173)(103,175)
(104,176)(105,178)(106,177)(107,179)(108,180);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope