Polytope of Type {4,6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6,3}*864c
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 5
Schlafli Type : {4,6,6,3}
Number of vertices, edges, etc : 4, 12, 18, 9, 3
Order of s0s1s2s3s4 : 3
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,6,3,2} of size 1728
Vertex Figure Of :
{2,4,6,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,3,6,3}*432
3-fold quotients : {4,6,2,3}*288b
6-fold quotients : {4,3,2,3}*144
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6,6,3}*1728a, {4,6,6,6}*1728c
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)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)( 18, 23)
( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 73)
( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)( 45, 77)
( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)( 53, 93)
( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)( 61, 97)
( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)
( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)(116,120)
(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)
(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)(150,191)
(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)(158,195)
(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)(166,199)
(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)(174,215)
(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);;
s2 := ( 1,181)( 2,184)( 3,183)( 4,182)( 5,189)( 6,192)( 7,191)( 8,190)
( 9,185)( 10,188)( 11,187)( 12,186)( 13,201)( 14,204)( 15,203)( 16,202)
( 17,197)( 18,200)( 19,199)( 20,198)( 21,193)( 22,196)( 23,195)( 24,194)
( 25,209)( 26,212)( 27,211)( 28,210)( 29,205)( 30,208)( 31,207)( 32,206)
( 33,213)( 34,216)( 35,215)( 36,214)( 37,145)( 38,148)( 39,147)( 40,146)
( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)( 48,150)
( 49,165)( 50,168)( 51,167)( 52,166)( 53,161)( 54,164)( 55,163)( 56,162)
( 57,157)( 58,160)( 59,159)( 60,158)( 61,173)( 62,176)( 63,175)( 64,174)
( 65,169)( 66,172)( 67,171)( 68,170)( 69,177)( 70,180)( 71,179)( 72,178)
( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)( 80,118)
( 81,113)( 82,116)( 83,115)( 84,114)( 85,129)( 86,132)( 87,131)( 88,130)
( 89,125)( 90,128)( 91,127)( 92,126)( 93,121)( 94,124)( 95,123)( 96,122)
( 97,137)( 98,140)( 99,139)(100,138)(101,133)(102,136)(103,135)(104,134)
(105,141)(106,144)(107,143)(108,142);;
s3 := ( 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);;
s4 := ( 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,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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s0*s1*s2*s0*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 ];;
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)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)
( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)
( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)
( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)
( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)
( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)
( 69,101)( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)
(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)
(138,143)(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)
(150,191)(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)
(158,195)(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)
(166,199)(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)
(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);
s2 := Sym(216)!( 1,181)( 2,184)( 3,183)( 4,182)( 5,189)( 6,192)( 7,191)
( 8,190)( 9,185)( 10,188)( 11,187)( 12,186)( 13,201)( 14,204)( 15,203)
( 16,202)( 17,197)( 18,200)( 19,199)( 20,198)( 21,193)( 22,196)( 23,195)
( 24,194)( 25,209)( 26,212)( 27,211)( 28,210)( 29,205)( 30,208)( 31,207)
( 32,206)( 33,213)( 34,216)( 35,215)( 36,214)( 37,145)( 38,148)( 39,147)
( 40,146)( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)
( 48,150)( 49,165)( 50,168)( 51,167)( 52,166)( 53,161)( 54,164)( 55,163)
( 56,162)( 57,157)( 58,160)( 59,159)( 60,158)( 61,173)( 62,176)( 63,175)
( 64,174)( 65,169)( 66,172)( 67,171)( 68,170)( 69,177)( 70,180)( 71,179)
( 72,178)( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)
( 80,118)( 81,113)( 82,116)( 83,115)( 84,114)( 85,129)( 86,132)( 87,131)
( 88,130)( 89,125)( 90,128)( 91,127)( 92,126)( 93,121)( 94,124)( 95,123)
( 96,122)( 97,137)( 98,140)( 99,139)(100,138)(101,133)(102,136)(103,135)
(104,134)(105,141)(106,144)(107,143)(108,142);
s3 := 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);
s4 := 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,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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 >;
References : None.
to this polytope