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