Polytope of Type {2,2,3,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,3,6,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47394)
Rank : 6
Schlafli Type : {2,2,3,6,12}
Number of vertices, edges, etc : 2, 2, 3, 9, 36, 12
Order of s0s1s2s3s4s5 : 12
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
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 : {2,2,3,6,6}*864b
3-fold quotients : {2,2,3,2,12}*576, {2,2,3,6,4}*576
6-fold quotients : {2,2,3,2,6}*288, {2,2,3,6,2}*288
9-fold quotients : {2,2,3,2,4}*192
12-fold quotients : {2,2,3,2,3}*144
18-fold quotients : {2,2,3,2,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 5,167)( 6,168)( 7,169)( 8,173)( 9,174)( 10,175)( 11,170)( 12,171)
( 13,172)( 14,185)( 15,186)( 16,187)( 17,191)( 18,192)( 19,193)( 20,188)
( 21,189)( 22,190)( 23,176)( 24,177)( 25,178)( 26,182)( 27,183)( 28,184)
( 29,179)( 30,180)( 31,181)( 32,194)( 33,195)( 34,196)( 35,200)( 36,201)
( 37,202)( 38,197)( 39,198)( 40,199)( 41,212)( 42,213)( 43,214)( 44,218)
( 45,219)( 46,220)( 47,215)( 48,216)( 49,217)( 50,203)( 51,204)( 52,205)
( 53,209)( 54,210)( 55,211)( 56,206)( 57,207)( 58,208)( 59,140)( 60,141)
( 61,142)( 62,146)( 63,147)( 64,148)( 65,143)( 66,144)( 67,145)( 68,158)
( 69,159)( 70,160)( 71,164)( 72,165)( 73,166)( 74,161)( 75,162)( 76,163)
( 77,149)( 78,150)( 79,151)( 80,155)( 81,156)( 82,157)( 83,152)( 84,153)
( 85,154)( 86,113)( 87,114)( 88,115)( 89,119)( 90,120)( 91,121)( 92,116)
( 93,117)( 94,118)( 95,131)( 96,132)( 97,133)( 98,137)( 99,138)(100,139)
(101,134)(102,135)(103,136)(104,122)(105,123)(106,124)(107,128)(108,129)
(109,130)(110,125)(111,126)(112,127);;
s3 := ( 5,179)( 6,180)( 7,181)( 8,176)( 9,177)( 10,178)( 11,182)( 12,183)
( 13,184)( 14,170)( 15,171)( 16,172)( 17,167)( 18,168)( 19,169)( 20,173)
( 21,174)( 22,175)( 23,188)( 24,189)( 25,190)( 26,185)( 27,186)( 28,187)
( 29,191)( 30,192)( 31,193)( 32,206)( 33,207)( 34,208)( 35,203)( 36,204)
( 37,205)( 38,209)( 39,210)( 40,211)( 41,197)( 42,198)( 43,199)( 44,194)
( 45,195)( 46,196)( 47,200)( 48,201)( 49,202)( 50,215)( 51,216)( 52,217)
( 53,212)( 54,213)( 55,214)( 56,218)( 57,219)( 58,220)( 59,152)( 60,153)
( 61,154)( 62,149)( 63,150)( 64,151)( 65,155)( 66,156)( 67,157)( 68,143)
( 69,144)( 70,145)( 71,140)( 72,141)( 73,142)( 74,146)( 75,147)( 76,148)
( 77,161)( 78,162)( 79,163)( 80,158)( 81,159)( 82,160)( 83,164)( 84,165)
( 85,166)( 86,125)( 87,126)( 88,127)( 89,122)( 90,123)( 91,124)( 92,128)
( 93,129)( 94,130)( 95,116)( 96,117)( 97,118)( 98,113)( 99,114)(100,115)
(101,119)(102,120)(103,121)(104,134)(105,135)(106,136)(107,131)(108,132)
(109,133)(110,137)(111,138)(112,139);;
s4 := ( 5,113)( 6,115)( 7,114)( 8,119)( 9,121)( 10,120)( 11,116)( 12,118)
( 13,117)( 14,122)( 15,124)( 16,123)( 17,128)( 18,130)( 19,129)( 20,125)
( 21,127)( 22,126)( 23,131)( 24,133)( 25,132)( 26,137)( 27,139)( 28,138)
( 29,134)( 30,136)( 31,135)( 32,140)( 33,142)( 34,141)( 35,146)( 36,148)
( 37,147)( 38,143)( 39,145)( 40,144)( 41,149)( 42,151)( 43,150)( 44,155)
( 45,157)( 46,156)( 47,152)( 48,154)( 49,153)( 50,158)( 51,160)( 52,159)
( 53,164)( 54,166)( 55,165)( 56,161)( 57,163)( 58,162)( 59,194)( 60,196)
( 61,195)( 62,200)( 63,202)( 64,201)( 65,197)( 66,199)( 67,198)( 68,203)
( 69,205)( 70,204)( 71,209)( 72,211)( 73,210)( 74,206)( 75,208)( 76,207)
( 77,212)( 78,214)( 79,213)( 80,218)( 81,220)( 82,219)( 83,215)( 84,217)
( 85,216)( 86,167)( 87,169)( 88,168)( 89,173)( 90,175)( 91,174)( 92,170)
( 93,172)( 94,171)( 95,176)( 96,178)( 97,177)( 98,182)( 99,184)(100,183)
(101,179)(102,181)(103,180)(104,185)(105,187)(106,186)(107,191)(108,193)
(109,192)(110,188)(111,190)(112,189);;
s5 := ( 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, 87)( 60, 86)( 61, 88)( 62, 90)( 63, 89)( 64, 91)
( 65, 93)( 66, 92)( 67, 94)( 68, 96)( 69, 95)( 70, 97)( 71, 99)( 72, 98)
( 73,100)( 74,102)( 75,101)( 76,103)( 77,105)( 78,104)( 79,106)( 80,108)
( 81,107)( 82,109)( 83,111)( 84,110)( 85,112)(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)(136,163)(137,165)(138,164)(139,166)
(167,168)(170,171)(173,174)(176,177)(179,180)(182,183)(185,186)(188,189)
(191,192)(194,195)(197,198)(200,201)(203,204)(206,207)(209,210)(212,213)
(215,216)(218,219);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s2*s3*s2*s3*s2*s3, s3*s4*s5*s4*s3*s4*s5*s4,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(220)!(1,2);
s1 := Sym(220)!(3,4);
s2 := Sym(220)!( 5,167)( 6,168)( 7,169)( 8,173)( 9,174)( 10,175)( 11,170)
( 12,171)( 13,172)( 14,185)( 15,186)( 16,187)( 17,191)( 18,192)( 19,193)
( 20,188)( 21,189)( 22,190)( 23,176)( 24,177)( 25,178)( 26,182)( 27,183)
( 28,184)( 29,179)( 30,180)( 31,181)( 32,194)( 33,195)( 34,196)( 35,200)
( 36,201)( 37,202)( 38,197)( 39,198)( 40,199)( 41,212)( 42,213)( 43,214)
( 44,218)( 45,219)( 46,220)( 47,215)( 48,216)( 49,217)( 50,203)( 51,204)
( 52,205)( 53,209)( 54,210)( 55,211)( 56,206)( 57,207)( 58,208)( 59,140)
( 60,141)( 61,142)( 62,146)( 63,147)( 64,148)( 65,143)( 66,144)( 67,145)
( 68,158)( 69,159)( 70,160)( 71,164)( 72,165)( 73,166)( 74,161)( 75,162)
( 76,163)( 77,149)( 78,150)( 79,151)( 80,155)( 81,156)( 82,157)( 83,152)
( 84,153)( 85,154)( 86,113)( 87,114)( 88,115)( 89,119)( 90,120)( 91,121)
( 92,116)( 93,117)( 94,118)( 95,131)( 96,132)( 97,133)( 98,137)( 99,138)
(100,139)(101,134)(102,135)(103,136)(104,122)(105,123)(106,124)(107,128)
(108,129)(109,130)(110,125)(111,126)(112,127);
s3 := Sym(220)!( 5,179)( 6,180)( 7,181)( 8,176)( 9,177)( 10,178)( 11,182)
( 12,183)( 13,184)( 14,170)( 15,171)( 16,172)( 17,167)( 18,168)( 19,169)
( 20,173)( 21,174)( 22,175)( 23,188)( 24,189)( 25,190)( 26,185)( 27,186)
( 28,187)( 29,191)( 30,192)( 31,193)( 32,206)( 33,207)( 34,208)( 35,203)
( 36,204)( 37,205)( 38,209)( 39,210)( 40,211)( 41,197)( 42,198)( 43,199)
( 44,194)( 45,195)( 46,196)( 47,200)( 48,201)( 49,202)( 50,215)( 51,216)
( 52,217)( 53,212)( 54,213)( 55,214)( 56,218)( 57,219)( 58,220)( 59,152)
( 60,153)( 61,154)( 62,149)( 63,150)( 64,151)( 65,155)( 66,156)( 67,157)
( 68,143)( 69,144)( 70,145)( 71,140)( 72,141)( 73,142)( 74,146)( 75,147)
( 76,148)( 77,161)( 78,162)( 79,163)( 80,158)( 81,159)( 82,160)( 83,164)
( 84,165)( 85,166)( 86,125)( 87,126)( 88,127)( 89,122)( 90,123)( 91,124)
( 92,128)( 93,129)( 94,130)( 95,116)( 96,117)( 97,118)( 98,113)( 99,114)
(100,115)(101,119)(102,120)(103,121)(104,134)(105,135)(106,136)(107,131)
(108,132)(109,133)(110,137)(111,138)(112,139);
s4 := Sym(220)!( 5,113)( 6,115)( 7,114)( 8,119)( 9,121)( 10,120)( 11,116)
( 12,118)( 13,117)( 14,122)( 15,124)( 16,123)( 17,128)( 18,130)( 19,129)
( 20,125)( 21,127)( 22,126)( 23,131)( 24,133)( 25,132)( 26,137)( 27,139)
( 28,138)( 29,134)( 30,136)( 31,135)( 32,140)( 33,142)( 34,141)( 35,146)
( 36,148)( 37,147)( 38,143)( 39,145)( 40,144)( 41,149)( 42,151)( 43,150)
( 44,155)( 45,157)( 46,156)( 47,152)( 48,154)( 49,153)( 50,158)( 51,160)
( 52,159)( 53,164)( 54,166)( 55,165)( 56,161)( 57,163)( 58,162)( 59,194)
( 60,196)( 61,195)( 62,200)( 63,202)( 64,201)( 65,197)( 66,199)( 67,198)
( 68,203)( 69,205)( 70,204)( 71,209)( 72,211)( 73,210)( 74,206)( 75,208)
( 76,207)( 77,212)( 78,214)( 79,213)( 80,218)( 81,220)( 82,219)( 83,215)
( 84,217)( 85,216)( 86,167)( 87,169)( 88,168)( 89,173)( 90,175)( 91,174)
( 92,170)( 93,172)( 94,171)( 95,176)( 96,178)( 97,177)( 98,182)( 99,184)
(100,183)(101,179)(102,181)(103,180)(104,185)(105,187)(106,186)(107,191)
(108,193)(109,192)(110,188)(111,190)(112,189);
s5 := Sym(220)!( 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, 87)( 60, 86)( 61, 88)( 62, 90)( 63, 89)
( 64, 91)( 65, 93)( 66, 92)( 67, 94)( 68, 96)( 69, 95)( 70, 97)( 71, 99)
( 72, 98)( 73,100)( 74,102)( 75,101)( 76,103)( 77,105)( 78,104)( 79,106)
( 80,108)( 81,107)( 82,109)( 83,111)( 84,110)( 85,112)(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)(136,163)(137,165)(138,164)
(139,166)(167,168)(170,171)(173,174)(176,177)(179,180)(182,183)(185,186)
(188,189)(191,192)(194,195)(197,198)(200,201)(203,204)(206,207)(209,210)
(212,213)(215,216)(218,219);
poly := sub<Sym(220)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5,
s3*s5*s3*s5, s2*s3*s2*s3*s2*s3, s3*s4*s5*s4*s3*s4*s5*s4,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;
to this polytope