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