Polytope of Type {12,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,6}*864
if this polytope has a name.
Group : SmallGroup(864,4080)
Rank : 4
Schlafli Type : {12,4,6}
Number of vertices, edges, etc : 12, 36, 18, 9
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,4,6,2} of size 1728
Vertex Figure Of :
{2,12,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4,6}*432b
3-fold quotients : {4,4,6}*288
6-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,4,6}*1728, {12,4,6}*1728a
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)(111,137)(112,139)
(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)(119,147)(120,146)
(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)(127,154)(128,156)
(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161)(163,190)
(164,192)(165,191)(166,193)(167,195)(168,194)(169,196)(170,198)(171,197)
(172,199)(173,201)(174,200)(175,202)(176,204)(177,203)(178,205)(179,207)
(180,206)(181,208)(182,210)(183,209)(184,211)(185,213)(186,212)(187,214)
(188,216)(189,215);;
s1 := ( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)
( 9,117)( 10,134)( 11,133)( 12,135)( 13,128)( 14,127)( 15,129)( 16,131)
( 17,130)( 18,132)( 19,122)( 20,121)( 21,123)( 22,125)( 23,124)( 24,126)
( 25,119)( 26,118)( 27,120)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)
( 33,141)( 34,143)( 35,142)( 36,144)( 37,161)( 38,160)( 39,162)( 40,155)
( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,149)( 47,148)( 48,150)
( 49,152)( 50,151)( 51,153)( 52,146)( 53,145)( 54,147)( 55,164)( 56,163)
( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)( 64,188)
( 65,187)( 66,189)( 67,182)( 68,181)( 69,183)( 70,185)( 71,184)( 72,186)
( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,173)( 80,172)
( 81,174)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)( 88,197)
( 89,196)( 90,198)( 91,215)( 92,214)( 93,216)( 94,209)( 95,208)( 96,210)
( 97,212)( 98,211)( 99,213)(100,203)(101,202)(102,204)(103,206)(104,205)
(105,207)(106,200)(107,199)(108,201);;
s2 := ( 4, 22)( 5, 23)( 6, 24)( 7, 16)( 8, 17)( 9, 18)( 10, 19)( 11, 20)
( 12, 21)( 31, 49)( 32, 50)( 33, 51)( 34, 43)( 35, 44)( 36, 45)( 37, 46)
( 38, 47)( 39, 48)( 58, 76)( 59, 77)( 60, 78)( 61, 70)( 62, 71)( 63, 72)
( 64, 73)( 65, 74)( 66, 75)( 85,103)( 86,104)( 87,105)( 88, 97)( 89, 98)
( 90, 99)( 91,100)( 92,101)( 93,102)(109,163)(110,164)(111,165)(112,184)
(113,185)(114,186)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)
(121,175)(122,176)(123,177)(124,169)(125,170)(126,171)(127,172)(128,173)
(129,174)(130,166)(131,167)(132,168)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,211)(140,212)(141,213)(142,205)(143,206)(144,207)
(145,208)(146,209)(147,210)(148,202)(149,203)(150,204)(151,196)(152,197)
(153,198)(154,199)(155,200)(156,201)(157,193)(158,194)(159,195)(160,214)
(161,215)(162,216);;
s3 := ( 1, 58)( 2, 59)( 3, 60)( 4, 55)( 5, 56)( 6, 57)( 7, 61)( 8, 62)
( 9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)( 16, 79)
( 17, 80)( 18, 81)( 19, 67)( 20, 68)( 21, 69)( 22, 64)( 23, 65)( 24, 66)
( 25, 70)( 26, 71)( 27, 72)( 28, 85)( 29, 86)( 30, 87)( 31, 82)( 32, 83)
( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)( 40,100)
( 41,101)( 42,102)( 43,106)( 44,107)( 45,108)( 46, 94)( 47, 95)( 48, 96)
( 49, 91)( 50, 92)( 51, 93)( 52, 97)( 53, 98)( 54, 99)(109,166)(110,167)
(111,168)(112,163)(113,164)(114,165)(115,169)(116,170)(117,171)(118,184)
(119,185)(120,186)(121,181)(122,182)(123,183)(124,187)(125,188)(126,189)
(127,175)(128,176)(129,177)(130,172)(131,173)(132,174)(133,178)(134,179)
(135,180)(136,193)(137,194)(138,195)(139,190)(140,191)(141,192)(142,196)
(143,197)(144,198)(145,211)(146,212)(147,213)(148,208)(149,209)(150,210)
(151,214)(152,215)(153,216)(154,202)(155,203)(156,204)(157,199)(158,200)
(159,201)(160,205)(161,206)(162,207);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*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)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)
( 71, 72)( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)(111,137)
(112,139)(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)(119,147)
(120,146)(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)(127,154)
(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161)
(163,190)(164,192)(165,191)(166,193)(167,195)(168,194)(169,196)(170,198)
(171,197)(172,199)(173,201)(174,200)(175,202)(176,204)(177,203)(178,205)
(179,207)(180,206)(181,208)(182,210)(183,209)(184,211)(185,213)(186,212)
(187,214)(188,216)(189,215);
s1 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)
( 8,115)( 9,117)( 10,134)( 11,133)( 12,135)( 13,128)( 14,127)( 15,129)
( 16,131)( 17,130)( 18,132)( 19,122)( 20,121)( 21,123)( 22,125)( 23,124)
( 24,126)( 25,119)( 26,118)( 27,120)( 28,137)( 29,136)( 30,138)( 31,140)
( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,161)( 38,160)( 39,162)
( 40,155)( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,149)( 47,148)
( 48,150)( 49,152)( 50,151)( 51,153)( 52,146)( 53,145)( 54,147)( 55,164)
( 56,163)( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)
( 64,188)( 65,187)( 66,189)( 67,182)( 68,181)( 69,183)( 70,185)( 71,184)
( 72,186)( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,173)
( 80,172)( 81,174)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)
( 88,197)( 89,196)( 90,198)( 91,215)( 92,214)( 93,216)( 94,209)( 95,208)
( 96,210)( 97,212)( 98,211)( 99,213)(100,203)(101,202)(102,204)(103,206)
(104,205)(105,207)(106,200)(107,199)(108,201);
s2 := Sym(216)!( 4, 22)( 5, 23)( 6, 24)( 7, 16)( 8, 17)( 9, 18)( 10, 19)
( 11, 20)( 12, 21)( 31, 49)( 32, 50)( 33, 51)( 34, 43)( 35, 44)( 36, 45)
( 37, 46)( 38, 47)( 39, 48)( 58, 76)( 59, 77)( 60, 78)( 61, 70)( 62, 71)
( 63, 72)( 64, 73)( 65, 74)( 66, 75)( 85,103)( 86,104)( 87,105)( 88, 97)
( 89, 98)( 90, 99)( 91,100)( 92,101)( 93,102)(109,163)(110,164)(111,165)
(112,184)(113,185)(114,186)(115,178)(116,179)(117,180)(118,181)(119,182)
(120,183)(121,175)(122,176)(123,177)(124,169)(125,170)(126,171)(127,172)
(128,173)(129,174)(130,166)(131,167)(132,168)(133,187)(134,188)(135,189)
(136,190)(137,191)(138,192)(139,211)(140,212)(141,213)(142,205)(143,206)
(144,207)(145,208)(146,209)(147,210)(148,202)(149,203)(150,204)(151,196)
(152,197)(153,198)(154,199)(155,200)(156,201)(157,193)(158,194)(159,195)
(160,214)(161,215)(162,216);
s3 := Sym(216)!( 1, 58)( 2, 59)( 3, 60)( 4, 55)( 5, 56)( 6, 57)( 7, 61)
( 8, 62)( 9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)
( 16, 79)( 17, 80)( 18, 81)( 19, 67)( 20, 68)( 21, 69)( 22, 64)( 23, 65)
( 24, 66)( 25, 70)( 26, 71)( 27, 72)( 28, 85)( 29, 86)( 30, 87)( 31, 82)
( 32, 83)( 33, 84)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)
( 40,100)( 41,101)( 42,102)( 43,106)( 44,107)( 45,108)( 46, 94)( 47, 95)
( 48, 96)( 49, 91)( 50, 92)( 51, 93)( 52, 97)( 53, 98)( 54, 99)(109,166)
(110,167)(111,168)(112,163)(113,164)(114,165)(115,169)(116,170)(117,171)
(118,184)(119,185)(120,186)(121,181)(122,182)(123,183)(124,187)(125,188)
(126,189)(127,175)(128,176)(129,177)(130,172)(131,173)(132,174)(133,178)
(134,179)(135,180)(136,193)(137,194)(138,195)(139,190)(140,191)(141,192)
(142,196)(143,197)(144,198)(145,211)(146,212)(147,213)(148,208)(149,209)
(150,210)(151,214)(152,215)(153,216)(154,202)(155,203)(156,204)(157,199)
(158,200)(159,201)(160,205)(161,206)(162,207);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope