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