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