Polytope of Type {2,12,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,6,3}*864b
if this polytope has a name.
Group : SmallGroup(864,4391)
Rank : 5
Schlafli Type : {2,12,6,3}
Number of vertices, edges, etc : 2, 12, 36, 9, 3
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,12,6,3,2} of size 1728
Vertex Figure Of :
{2,2,12,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,6,3}*432b
3-fold quotients : {2,12,2,3}*288, {2,4,6,3}*288
6-fold quotients : {2,2,6,3}*144, {2,6,2,3}*144
9-fold quotients : {2,4,2,3}*96
12-fold quotients : {2,3,2,3}*72
18-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,24,6,3}*1728b, {4,12,6,3}*1728d, {2,12,6,6}*1728e
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)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)
( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 95)( 68, 94)( 69, 96)( 70, 98)
( 71, 97)( 72, 99)( 73,101)( 74,100)( 75,102)( 76,104)( 77,103)( 78,105)
( 79,107)( 80,106)( 81,108)( 82,110)( 83,109)(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)
(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)(184,185)(187,188)
(190,191)(193,194)(196,197)(199,200)(202,203)(205,206)(208,209)(211,212)
(214,215)(217,218);;
s2 := ( 3,112)( 4,111)( 5,113)( 6,118)( 7,117)( 8,119)( 9,115)( 10,114)
( 11,116)( 12,121)( 13,120)( 14,122)( 15,127)( 16,126)( 17,128)( 18,124)
( 19,123)( 20,125)( 21,130)( 22,129)( 23,131)( 24,136)( 25,135)( 26,137)
( 27,133)( 28,132)( 29,134)( 30,139)( 31,138)( 32,140)( 33,145)( 34,144)
( 35,146)( 36,142)( 37,141)( 38,143)( 39,148)( 40,147)( 41,149)( 42,154)
( 43,153)( 44,155)( 45,151)( 46,150)( 47,152)( 48,157)( 49,156)( 50,158)
( 51,163)( 52,162)( 53,164)( 54,160)( 55,159)( 56,161)( 57,193)( 58,192)
( 59,194)( 60,199)( 61,198)( 62,200)( 63,196)( 64,195)( 65,197)( 66,202)
( 67,201)( 68,203)( 69,208)( 70,207)( 71,209)( 72,205)( 73,204)( 74,206)
( 75,211)( 76,210)( 77,212)( 78,217)( 79,216)( 80,218)( 81,214)( 82,213)
( 83,215)( 84,166)( 85,165)( 86,167)( 87,172)( 88,171)( 89,173)( 90,169)
( 91,168)( 92,170)( 93,175)( 94,174)( 95,176)( 96,181)( 97,180)( 98,182)
( 99,178)(100,177)(101,179)(102,184)(103,183)(104,185)(105,190)(106,189)
(107,191)(108,187)(109,186)(110,188);;
s3 := ( 3,168)( 4,169)( 5,170)( 6,165)( 7,166)( 8,167)( 9,171)( 10,172)
( 11,173)( 12,186)( 13,187)( 14,188)( 15,183)( 16,184)( 17,185)( 18,189)
( 19,190)( 20,191)( 21,177)( 22,178)( 23,179)( 24,174)( 25,175)( 26,176)
( 27,180)( 28,181)( 29,182)( 30,195)( 31,196)( 32,197)( 33,192)( 34,193)
( 35,194)( 36,198)( 37,199)( 38,200)( 39,213)( 40,214)( 41,215)( 42,210)
( 43,211)( 44,212)( 45,216)( 46,217)( 47,218)( 48,204)( 49,205)( 50,206)
( 51,201)( 52,202)( 53,203)( 54,207)( 55,208)( 56,209)( 57,141)( 58,142)
( 59,143)( 60,138)( 61,139)( 62,140)( 63,144)( 64,145)( 65,146)( 66,159)
( 67,160)( 68,161)( 69,156)( 70,157)( 71,158)( 72,162)( 73,163)( 74,164)
( 75,150)( 76,151)( 77,152)( 78,147)( 79,148)( 80,149)( 81,153)( 82,154)
( 83,155)( 84,114)( 85,115)( 86,116)( 87,111)( 88,112)( 89,113)( 90,117)
( 91,118)( 92,119)( 93,132)( 94,133)( 95,134)( 96,129)( 97,130)( 98,131)
( 99,135)(100,136)(101,137)(102,123)(103,124)(104,125)(105,120)(106,121)
(107,122)(108,126)(109,127)(110,128);;
s4 := ( 3,174)( 4,175)( 5,176)( 6,180)( 7,181)( 8,182)( 9,177)( 10,178)
( 11,179)( 12,165)( 13,166)( 14,167)( 15,171)( 16,172)( 17,173)( 18,168)
( 19,169)( 20,170)( 21,183)( 22,184)( 23,185)( 24,189)( 25,190)( 26,191)
( 27,186)( 28,187)( 29,188)( 30,201)( 31,202)( 32,203)( 33,207)( 34,208)
( 35,209)( 36,204)( 37,205)( 38,206)( 39,192)( 40,193)( 41,194)( 42,198)
( 43,199)( 44,200)( 45,195)( 46,196)( 47,197)( 48,210)( 49,211)( 50,212)
( 51,216)( 52,217)( 53,218)( 54,213)( 55,214)( 56,215)( 57,147)( 58,148)
( 59,149)( 60,153)( 61,154)( 62,155)( 63,150)( 64,151)( 65,152)( 66,138)
( 67,139)( 68,140)( 69,144)( 70,145)( 71,146)( 72,141)( 73,142)( 74,143)
( 75,156)( 76,157)( 77,158)( 78,162)( 79,163)( 80,164)( 81,159)( 82,160)
( 83,161)( 84,120)( 85,121)( 86,122)( 87,126)( 88,127)( 89,128)( 90,123)
( 91,124)( 92,125)( 93,111)( 94,112)( 95,113)( 96,117)( 97,118)( 98,119)
( 99,114)(100,115)(101,116)(102,129)(103,130)(104,131)(105,135)(106,136)
(107,137)(108,132)(109,133)(110,134);;
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,
s3*s4*s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 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)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)
( 62, 88)( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 95)( 68, 94)( 69, 96)
( 70, 98)( 71, 97)( 72, 99)( 73,101)( 74,100)( 75,102)( 76,104)( 77,103)
( 78,105)( 79,107)( 80,106)( 81,108)( 82,110)( 83,109)(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)(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)(184,185)
(187,188)(190,191)(193,194)(196,197)(199,200)(202,203)(205,206)(208,209)
(211,212)(214,215)(217,218);
s2 := Sym(218)!( 3,112)( 4,111)( 5,113)( 6,118)( 7,117)( 8,119)( 9,115)
( 10,114)( 11,116)( 12,121)( 13,120)( 14,122)( 15,127)( 16,126)( 17,128)
( 18,124)( 19,123)( 20,125)( 21,130)( 22,129)( 23,131)( 24,136)( 25,135)
( 26,137)( 27,133)( 28,132)( 29,134)( 30,139)( 31,138)( 32,140)( 33,145)
( 34,144)( 35,146)( 36,142)( 37,141)( 38,143)( 39,148)( 40,147)( 41,149)
( 42,154)( 43,153)( 44,155)( 45,151)( 46,150)( 47,152)( 48,157)( 49,156)
( 50,158)( 51,163)( 52,162)( 53,164)( 54,160)( 55,159)( 56,161)( 57,193)
( 58,192)( 59,194)( 60,199)( 61,198)( 62,200)( 63,196)( 64,195)( 65,197)
( 66,202)( 67,201)( 68,203)( 69,208)( 70,207)( 71,209)( 72,205)( 73,204)
( 74,206)( 75,211)( 76,210)( 77,212)( 78,217)( 79,216)( 80,218)( 81,214)
( 82,213)( 83,215)( 84,166)( 85,165)( 86,167)( 87,172)( 88,171)( 89,173)
( 90,169)( 91,168)( 92,170)( 93,175)( 94,174)( 95,176)( 96,181)( 97,180)
( 98,182)( 99,178)(100,177)(101,179)(102,184)(103,183)(104,185)(105,190)
(106,189)(107,191)(108,187)(109,186)(110,188);
s3 := Sym(218)!( 3,168)( 4,169)( 5,170)( 6,165)( 7,166)( 8,167)( 9,171)
( 10,172)( 11,173)( 12,186)( 13,187)( 14,188)( 15,183)( 16,184)( 17,185)
( 18,189)( 19,190)( 20,191)( 21,177)( 22,178)( 23,179)( 24,174)( 25,175)
( 26,176)( 27,180)( 28,181)( 29,182)( 30,195)( 31,196)( 32,197)( 33,192)
( 34,193)( 35,194)( 36,198)( 37,199)( 38,200)( 39,213)( 40,214)( 41,215)
( 42,210)( 43,211)( 44,212)( 45,216)( 46,217)( 47,218)( 48,204)( 49,205)
( 50,206)( 51,201)( 52,202)( 53,203)( 54,207)( 55,208)( 56,209)( 57,141)
( 58,142)( 59,143)( 60,138)( 61,139)( 62,140)( 63,144)( 64,145)( 65,146)
( 66,159)( 67,160)( 68,161)( 69,156)( 70,157)( 71,158)( 72,162)( 73,163)
( 74,164)( 75,150)( 76,151)( 77,152)( 78,147)( 79,148)( 80,149)( 81,153)
( 82,154)( 83,155)( 84,114)( 85,115)( 86,116)( 87,111)( 88,112)( 89,113)
( 90,117)( 91,118)( 92,119)( 93,132)( 94,133)( 95,134)( 96,129)( 97,130)
( 98,131)( 99,135)(100,136)(101,137)(102,123)(103,124)(104,125)(105,120)
(106,121)(107,122)(108,126)(109,127)(110,128);
s4 := Sym(218)!( 3,174)( 4,175)( 5,176)( 6,180)( 7,181)( 8,182)( 9,177)
( 10,178)( 11,179)( 12,165)( 13,166)( 14,167)( 15,171)( 16,172)( 17,173)
( 18,168)( 19,169)( 20,170)( 21,183)( 22,184)( 23,185)( 24,189)( 25,190)
( 26,191)( 27,186)( 28,187)( 29,188)( 30,201)( 31,202)( 32,203)( 33,207)
( 34,208)( 35,209)( 36,204)( 37,205)( 38,206)( 39,192)( 40,193)( 41,194)
( 42,198)( 43,199)( 44,200)( 45,195)( 46,196)( 47,197)( 48,210)( 49,211)
( 50,212)( 51,216)( 52,217)( 53,218)( 54,213)( 55,214)( 56,215)( 57,147)
( 58,148)( 59,149)( 60,153)( 61,154)( 62,155)( 63,150)( 64,151)( 65,152)
( 66,138)( 67,139)( 68,140)( 69,144)( 70,145)( 71,146)( 72,141)( 73,142)
( 74,143)( 75,156)( 76,157)( 77,158)( 78,162)( 79,163)( 80,164)( 81,159)
( 82,160)( 83,161)( 84,120)( 85,121)( 86,122)( 87,126)( 88,127)( 89,128)
( 90,123)( 91,124)( 92,125)( 93,111)( 94,112)( 95,113)( 96,117)( 97,118)
( 98,119)( 99,114)(100,115)(101,116)(102,129)(103,130)(104,131)(105,135)
(106,136)(107,137)(108,132)(109,133)(110,134);
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, s3*s4*s3*s4*s3*s4,
s1*s2*s3*s2*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
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