Polytope of Type {2,12,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,4,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,46611)
Rank : 5
Schlafli Type : {2,12,4,6}
Number of vertices, edges, etc : 2, 12, 36, 18, 9
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,6,4,6}*864b
3-fold quotients : {2,4,4,6}*576
6-fold quotients : {2,2,4,6}*288
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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)(109,110)(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)(136,164)(137,163)(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)
(190,218)(191,217);;
s2 := ( 3,112)( 4,111)( 5,113)( 6,115)( 7,114)( 8,116)( 9,118)( 10,117)
( 11,119)( 12,136)( 13,135)( 14,137)( 15,130)( 16,129)( 17,131)( 18,133)
( 19,132)( 20,134)( 21,124)( 22,123)( 23,125)( 24,127)( 25,126)( 26,128)
( 27,121)( 28,120)( 29,122)( 30,139)( 31,138)( 32,140)( 33,142)( 34,141)
( 35,143)( 36,145)( 37,144)( 38,146)( 39,163)( 40,162)( 41,164)( 42,157)
( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,151)( 49,150)( 50,152)
( 51,154)( 52,153)( 53,155)( 54,148)( 55,147)( 56,149)( 57,166)( 58,165)
( 59,167)( 60,169)( 61,168)( 62,170)( 63,172)( 64,171)( 65,173)( 66,190)
( 67,189)( 68,191)( 69,184)( 70,183)( 71,185)( 72,187)( 73,186)( 74,188)
( 75,178)( 76,177)( 77,179)( 78,181)( 79,180)( 80,182)( 81,175)( 82,174)
( 83,176)( 84,193)( 85,192)( 86,194)( 87,196)( 88,195)( 89,197)( 90,199)
( 91,198)( 92,200)( 93,217)( 94,216)( 95,218)( 96,211)( 97,210)( 98,212)
( 99,214)(100,213)(101,215)(102,205)(103,204)(104,206)(105,208)(106,207)
(107,209)(108,202)(109,201)(110,203);;
s3 := ( 6, 24)( 7, 25)( 8, 26)( 9, 18)( 10, 19)( 11, 20)( 12, 21)( 13, 22)
( 14, 23)( 33, 51)( 34, 52)( 35, 53)( 36, 45)( 37, 46)( 38, 47)( 39, 48)
( 40, 49)( 41, 50)( 60, 78)( 61, 79)( 62, 80)( 63, 72)( 64, 73)( 65, 74)
( 66, 75)( 67, 76)( 68, 77)( 87,105)( 88,106)( 89,107)( 90, 99)( 91,100)
( 92,101)( 93,102)( 94,103)( 95,104)(111,165)(112,166)(113,167)(114,186)
(115,187)(116,188)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)
(123,177)(124,178)(125,179)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,168)(133,169)(134,170)(135,189)(136,190)(137,191)(138,192)
(139,193)(140,194)(141,213)(142,214)(143,215)(144,207)(145,208)(146,209)
(147,210)(148,211)(149,212)(150,204)(151,205)(152,206)(153,198)(154,199)
(155,200)(156,201)(157,202)(158,203)(159,195)(160,196)(161,197)(162,216)
(163,217)(164,218);;
s4 := ( 3, 60)( 4, 61)( 5, 62)( 6, 57)( 7, 58)( 8, 59)( 9, 63)( 10, 64)
( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 75)( 16, 76)( 17, 77)( 18, 81)
( 19, 82)( 20, 83)( 21, 69)( 22, 70)( 23, 71)( 24, 66)( 25, 67)( 26, 68)
( 27, 72)( 28, 73)( 29, 74)( 30, 87)( 31, 88)( 32, 89)( 33, 84)( 34, 85)
( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)( 42,102)
( 43,103)( 44,104)( 45,108)( 46,109)( 47,110)( 48, 96)( 49, 97)( 50, 98)
( 51, 93)( 52, 94)( 53, 95)( 54, 99)( 55,100)( 56,101)(111,168)(112,169)
(113,170)(114,165)(115,166)(116,167)(117,171)(118,172)(119,173)(120,186)
(121,187)(122,188)(123,183)(124,184)(125,185)(126,189)(127,190)(128,191)
(129,177)(130,178)(131,179)(132,174)(133,175)(134,176)(135,180)(136,181)
(137,182)(138,195)(139,196)(140,197)(141,192)(142,193)(143,194)(144,198)
(145,199)(146,200)(147,213)(148,214)(149,215)(150,210)(151,211)(152,212)
(153,216)(154,217)(155,218)(156,204)(157,205)(158,206)(159,201)(160,202)
(161,203)(162,207)(163,208)(164,209);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s3*s4*s3*s2*s3*s4*s2*s3*s2*s3*s4*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 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)(109,110)(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)(136,164)(137,163)
(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)(190,218)(191,217);
s2 := Sym(218)!( 3,112)( 4,111)( 5,113)( 6,115)( 7,114)( 8,116)( 9,118)
( 10,117)( 11,119)( 12,136)( 13,135)( 14,137)( 15,130)( 16,129)( 17,131)
( 18,133)( 19,132)( 20,134)( 21,124)( 22,123)( 23,125)( 24,127)( 25,126)
( 26,128)( 27,121)( 28,120)( 29,122)( 30,139)( 31,138)( 32,140)( 33,142)
( 34,141)( 35,143)( 36,145)( 37,144)( 38,146)( 39,163)( 40,162)( 41,164)
( 42,157)( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,151)( 49,150)
( 50,152)( 51,154)( 52,153)( 53,155)( 54,148)( 55,147)( 56,149)( 57,166)
( 58,165)( 59,167)( 60,169)( 61,168)( 62,170)( 63,172)( 64,171)( 65,173)
( 66,190)( 67,189)( 68,191)( 69,184)( 70,183)( 71,185)( 72,187)( 73,186)
( 74,188)( 75,178)( 76,177)( 77,179)( 78,181)( 79,180)( 80,182)( 81,175)
( 82,174)( 83,176)( 84,193)( 85,192)( 86,194)( 87,196)( 88,195)( 89,197)
( 90,199)( 91,198)( 92,200)( 93,217)( 94,216)( 95,218)( 96,211)( 97,210)
( 98,212)( 99,214)(100,213)(101,215)(102,205)(103,204)(104,206)(105,208)
(106,207)(107,209)(108,202)(109,201)(110,203);
s3 := Sym(218)!( 6, 24)( 7, 25)( 8, 26)( 9, 18)( 10, 19)( 11, 20)( 12, 21)
( 13, 22)( 14, 23)( 33, 51)( 34, 52)( 35, 53)( 36, 45)( 37, 46)( 38, 47)
( 39, 48)( 40, 49)( 41, 50)( 60, 78)( 61, 79)( 62, 80)( 63, 72)( 64, 73)
( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 87,105)( 88,106)( 89,107)( 90, 99)
( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)(111,165)(112,166)(113,167)
(114,186)(115,187)(116,188)(117,180)(118,181)(119,182)(120,183)(121,184)
(122,185)(123,177)(124,178)(125,179)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,168)(133,169)(134,170)(135,189)(136,190)(137,191)
(138,192)(139,193)(140,194)(141,213)(142,214)(143,215)(144,207)(145,208)
(146,209)(147,210)(148,211)(149,212)(150,204)(151,205)(152,206)(153,198)
(154,199)(155,200)(156,201)(157,202)(158,203)(159,195)(160,196)(161,197)
(162,216)(163,217)(164,218);
s4 := Sym(218)!( 3, 60)( 4, 61)( 5, 62)( 6, 57)( 7, 58)( 8, 59)( 9, 63)
( 10, 64)( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 75)( 16, 76)( 17, 77)
( 18, 81)( 19, 82)( 20, 83)( 21, 69)( 22, 70)( 23, 71)( 24, 66)( 25, 67)
( 26, 68)( 27, 72)( 28, 73)( 29, 74)( 30, 87)( 31, 88)( 32, 89)( 33, 84)
( 34, 85)( 35, 86)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)
( 42,102)( 43,103)( 44,104)( 45,108)( 46,109)( 47,110)( 48, 96)( 49, 97)
( 50, 98)( 51, 93)( 52, 94)( 53, 95)( 54, 99)( 55,100)( 56,101)(111,168)
(112,169)(113,170)(114,165)(115,166)(116,167)(117,171)(118,172)(119,173)
(120,186)(121,187)(122,188)(123,183)(124,184)(125,185)(126,189)(127,190)
(128,191)(129,177)(130,178)(131,179)(132,174)(133,175)(134,176)(135,180)
(136,181)(137,182)(138,195)(139,196)(140,197)(141,192)(142,193)(143,194)
(144,198)(145,199)(146,200)(147,213)(148,214)(149,215)(150,210)(151,211)
(152,212)(153,216)(154,217)(155,218)(156,204)(157,205)(158,206)(159,201)
(160,202)(161,203)(162,207)(163,208)(164,209);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s3*s4*s3*s2*s3*s4*s2*s3*s2*s3*s4*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope