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