Polytope of Type {4,3,6,6}

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