Polytope of Type {4,6,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6,6}*1728l
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {4,6,6,6}
Number of vertices, edges, etc : 4, 12, 18, 18, 6
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Non-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 : {4,6,6,2}*576d
9-fold quotients : {4,6,2,2}*192c
18-fold quotients : {4,3,2,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
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 := ( 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);;
s2 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)( 8, 46)
( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)( 32, 58)
( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 74, 76)( 77, 81)( 78, 84)( 79, 83)
( 80, 82)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89,105)( 90,108)( 91,107)
( 92,106)( 93,101)( 94,104)( 95,103)( 96,102)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)(127,179)
(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)(135,159)
(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)(143,163)
(144,162)(182,184)(185,189)(186,192)(187,191)(188,190)(193,205)(194,208)
(195,207)(196,206)(197,213)(198,216)(199,215)(200,214)(201,209)(202,212)
(203,211)(204,210);;
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*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)!( 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)!( 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);
s2 := Sym(216)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)
( 8, 46)( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)
( 24, 66)( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)
( 32, 58)( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 74, 76)( 77, 81)( 78, 84)
( 79, 83)( 80, 82)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89,105)( 90,108)
( 91,107)( 92,106)( 93,101)( 94,104)( 95,103)( 96,102)(109,145)(110,148)
(111,147)(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)
(119,151)(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)
(127,179)(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)
(135,159)(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)
(143,163)(144,162)(182,184)(185,189)(186,192)(187,191)(188,190)(193,205)
(194,208)(195,207)(196,206)(197,213)(198,216)(199,215)(200,214)(201,209)
(202,212)(203,211)(204,210);
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*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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