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}*1728d
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {6,6,6,3}
Number of vertices, edges, etc : 8, 24, 24, 9, 3
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) :
2-fold quotients : {3,6,6,3}*864b
3-fold quotients : {6,6,2,3}*576
6-fold quotients : {3,6,2,3}*288, {6,3,2,3}*288
12-fold quotients : {3,3,2,3}*144, {2,2,6,3}*144
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 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);;
s2 := ( 1, 2)( 5, 6)( 9, 10)( 13, 26)( 14, 25)( 15, 27)( 16, 28)( 17, 30)
( 18, 29)( 19, 31)( 20, 32)( 21, 34)( 22, 33)( 23, 35)( 24, 36)( 37, 74)
( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)( 44, 80)( 45, 82)
( 46, 81)( 47, 83)( 48, 84)( 49, 98)( 50, 97)( 51, 99)( 52,100)( 53,102)
( 54,101)( 55,103)( 56,104)( 57,106)( 58,105)( 59,107)( 60,108)( 61, 86)
( 62, 85)( 63, 87)( 64, 88)( 65, 90)( 66, 89)( 67, 91)( 68, 92)( 69, 94)
( 70, 93)( 71, 95)( 72, 96)(109,110)(113,114)(117,118)(121,134)(122,133)
(123,135)(124,136)(125,138)(126,137)(127,139)(128,140)(129,142)(130,141)
(131,143)(132,144)(145,182)(146,181)(147,183)(148,184)(149,186)(150,185)
(151,187)(152,188)(153,190)(154,189)(155,191)(156,192)(157,206)(158,205)
(159,207)(160,208)(161,210)(162,209)(163,211)(164,212)(165,214)(166,213)
(167,215)(168,216)(169,194)(170,193)(171,195)(172,196)(173,198)(174,197)
(175,199)(176,200)(177,202)(178,201)(179,203)(180,204);;
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, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 29)( 14, 30)( 15, 31)( 16, 32)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)( 24, 36)
( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 65)( 50, 66)( 51, 67)( 52, 68)
( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85,101)( 86,102)( 87,103)( 88,104)
( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)( 96,108)
(109,113)(110,114)(111,115)(112,116)(121,137)(122,138)(123,139)(124,140)
(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)(132,144)
(145,149)(146,150)(147,151)(148,152)(157,173)(158,174)(159,175)(160,176)
(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)(168,180)
(181,185)(182,186)(183,187)(184,188)(193,209)(194,210)(195,211)(196,212)
(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)(204,216);;
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, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := 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);
s2 := Sym(216)!( 1, 2)( 5, 6)( 9, 10)( 13, 26)( 14, 25)( 15, 27)( 16, 28)
( 17, 30)( 18, 29)( 19, 31)( 20, 32)( 21, 34)( 22, 33)( 23, 35)( 24, 36)
( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)( 44, 80)
( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 98)( 50, 97)( 51, 99)( 52,100)
( 53,102)( 54,101)( 55,103)( 56,104)( 57,106)( 58,105)( 59,107)( 60,108)
( 61, 86)( 62, 85)( 63, 87)( 64, 88)( 65, 90)( 66, 89)( 67, 91)( 68, 92)
( 69, 94)( 70, 93)( 71, 95)( 72, 96)(109,110)(113,114)(117,118)(121,134)
(122,133)(123,135)(124,136)(125,138)(126,137)(127,139)(128,140)(129,142)
(130,141)(131,143)(132,144)(145,182)(146,181)(147,183)(148,184)(149,186)
(150,185)(151,187)(152,188)(153,190)(154,189)(155,191)(156,192)(157,206)
(158,205)(159,207)(160,208)(161,210)(162,209)(163,211)(164,212)(165,214)
(166,213)(167,215)(168,216)(169,194)(170,193)(171,195)(172,196)(173,198)
(174,197)(175,199)(176,200)(177,202)(178,201)(179,203)(180,204);
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, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 29)( 14, 30)( 15, 31)
( 16, 32)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)
( 24, 36)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 65)( 50, 66)( 51, 67)
( 52, 68)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85,101)( 86,102)( 87,103)
( 88,104)( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)
( 96,108)(109,113)(110,114)(111,115)(112,116)(121,137)(122,138)(123,139)
(124,140)(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)
(132,144)(145,149)(146,150)(147,151)(148,152)(157,173)(158,174)(159,175)
(160,176)(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)
(168,180)(181,185)(182,186)(183,187)(184,188)(193,209)(194,210)(195,211)
(196,212)(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)
(204,216);
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,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope