Polytope of Type {6,12,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,8}*1728c
if this polytope has a name.
Group : SmallGroup(1728,33596)
Rank : 4
Schlafli Type : {6,12,8}
Number of vertices, edges, etc : 9, 54, 72, 8
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 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) :
2-fold quotients : {6,12,4}*864c
3-fold quotients : {6,4,8}*576
4-fold quotients : {6,12,2}*432c
6-fold quotients : {6,4,4}*288
12-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)
(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)
(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)
(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)
(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)
(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);;
s1 := ( 1, 4)( 3, 9)( 5, 8)( 10, 22)( 11, 20)( 12, 27)( 13, 19)( 14, 26)
( 15, 24)( 16, 25)( 17, 23)( 18, 21)( 28, 31)( 30, 36)( 32, 35)( 37, 49)
( 38, 47)( 39, 54)( 40, 46)( 41, 53)( 42, 51)( 43, 52)( 44, 50)( 45, 48)
( 55, 58)( 57, 63)( 59, 62)( 64, 76)( 65, 74)( 66, 81)( 67, 73)( 68, 80)
( 69, 78)( 70, 79)( 71, 77)( 72, 75)( 82, 85)( 84, 90)( 86, 89)( 91,103)
( 92,101)( 93,108)( 94,100)( 95,107)( 96,105)( 97,106)( 98,104)( 99,102)
(109,112)(111,117)(113,116)(118,130)(119,128)(120,135)(121,127)(122,134)
(123,132)(124,133)(125,131)(126,129)(136,139)(138,144)(140,143)(145,157)
(146,155)(147,162)(148,154)(149,161)(150,159)(151,160)(152,158)(153,156)
(163,166)(165,171)(167,170)(172,184)(173,182)(174,189)(175,181)(176,188)
(177,186)(178,187)(179,185)(180,183)(190,193)(192,198)(194,197)(199,211)
(200,209)(201,216)(202,208)(203,215)(204,213)(205,214)(206,212)(207,210);;
s2 := ( 1, 10)( 2, 11)( 3, 12)( 4, 18)( 5, 16)( 6, 17)( 7, 14)( 8, 15)
( 9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 91)( 56, 92)( 57, 93)( 58, 99)( 59, 97)( 60, 98)( 61, 95)( 62, 96)
( 63, 94)( 64, 82)( 65, 83)( 66, 84)( 67, 90)( 68, 88)( 69, 89)( 70, 86)
( 71, 87)( 72, 85)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)
( 79,104)( 80,105)( 81,103)(109,172)(110,173)(111,174)(112,180)(113,178)
(114,179)(115,176)(116,177)(117,175)(118,163)(119,164)(120,165)(121,171)
(122,169)(123,170)(124,167)(125,168)(126,166)(127,181)(128,182)(129,183)
(130,189)(131,187)(132,188)(133,185)(134,186)(135,184)(136,199)(137,200)
(138,201)(139,207)(140,205)(141,206)(142,203)(143,204)(144,202)(145,190)
(146,191)(147,192)(148,198)(149,196)(150,197)(151,194)(152,195)(153,193)
(154,208)(155,209)(156,210)(157,216)(158,214)(159,215)(160,212)(161,213)
(162,211);;
s3 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)
(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)
(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)
(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)
(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)
(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)
(213,215);
s1 := Sym(216)!( 1, 4)( 3, 9)( 5, 8)( 10, 22)( 11, 20)( 12, 27)( 13, 19)
( 14, 26)( 15, 24)( 16, 25)( 17, 23)( 18, 21)( 28, 31)( 30, 36)( 32, 35)
( 37, 49)( 38, 47)( 39, 54)( 40, 46)( 41, 53)( 42, 51)( 43, 52)( 44, 50)
( 45, 48)( 55, 58)( 57, 63)( 59, 62)( 64, 76)( 65, 74)( 66, 81)( 67, 73)
( 68, 80)( 69, 78)( 70, 79)( 71, 77)( 72, 75)( 82, 85)( 84, 90)( 86, 89)
( 91,103)( 92,101)( 93,108)( 94,100)( 95,107)( 96,105)( 97,106)( 98,104)
( 99,102)(109,112)(111,117)(113,116)(118,130)(119,128)(120,135)(121,127)
(122,134)(123,132)(124,133)(125,131)(126,129)(136,139)(138,144)(140,143)
(145,157)(146,155)(147,162)(148,154)(149,161)(150,159)(151,160)(152,158)
(153,156)(163,166)(165,171)(167,170)(172,184)(173,182)(174,189)(175,181)
(176,188)(177,186)(178,187)(179,185)(180,183)(190,193)(192,198)(194,197)
(199,211)(200,209)(201,216)(202,208)(203,215)(204,213)(205,214)(206,212)
(207,210);
s2 := Sym(216)!( 1, 10)( 2, 11)( 3, 12)( 4, 18)( 5, 16)( 6, 17)( 7, 14)
( 8, 15)( 9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)
( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)
( 51, 53)( 55, 91)( 56, 92)( 57, 93)( 58, 99)( 59, 97)( 60, 98)( 61, 95)
( 62, 96)( 63, 94)( 64, 82)( 65, 83)( 66, 84)( 67, 90)( 68, 88)( 69, 89)
( 70, 86)( 71, 87)( 72, 85)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)
( 78,107)( 79,104)( 80,105)( 81,103)(109,172)(110,173)(111,174)(112,180)
(113,178)(114,179)(115,176)(116,177)(117,175)(118,163)(119,164)(120,165)
(121,171)(122,169)(123,170)(124,167)(125,168)(126,166)(127,181)(128,182)
(129,183)(130,189)(131,187)(132,188)(133,185)(134,186)(135,184)(136,199)
(137,200)(138,201)(139,207)(140,205)(141,206)(142,203)(143,204)(144,202)
(145,190)(146,191)(147,192)(148,198)(149,196)(150,197)(151,194)(152,195)
(153,193)(154,208)(155,209)(156,210)(157,216)(158,214)(159,215)(160,212)
(161,213)(162,211);
s3 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope