Polytope of Type {12,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,3}*432b
if this polytope has a name.
Group : SmallGroup(432,602)
Rank : 4
Schlafli Type : {12,6,3}
Number of vertices, edges, etc : 12, 36, 9, 3
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,6,3,2} of size 864
   {12,6,3,4} of size 1728
Vertex Figure Of :
   {2,12,6,3} of size 864
   {4,12,6,3} of size 1728
   {3,12,6,3} of size 1728
   {4,12,6,3} of size 1728
   {4,12,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,3}*216b
   3-fold quotients : {12,2,3}*144, {4,6,3}*144
   6-fold quotients : {2,6,3}*72, {6,2,3}*72
   9-fold quotients : {4,2,3}*48
   12-fold quotients : {3,2,3}*36
   18-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6,3}*864b, {12,6,6}*864e
   3-fold covers : {36,6,3}*1296b, {12,6,9}*1296b, {12,6,3}*1296c, {12,6,3}*1296d, {12,6,3}*1296e
   4-fold covers : {48,6,3}*1728b, {24,6,6}*1728e, {12,6,12}*1728e, {12,12,6}*1728g, {12,6,3}*1728, {12,12,3}*1728b
Permutation Representation (GAP) :

s0 := (  2,  3)(  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)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 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)(109,136)(110,138)(111,137)
(112,139)(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)
(164,165)(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);;
s1 := (  1,110)(  2,109)(  3,111)(  4,116)(  5,115)(  6,117)(  7,113)(  8,112)
(  9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)
( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)( 24,135)
( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)
( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)( 40,152)
( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)( 48,156)
( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,191)( 56,190)
( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)( 64,200)
( 65,199)( 66,201)( 67,206)( 68,205)( 69,207)( 70,203)( 71,202)( 72,204)
( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)( 80,211)
( 81,213)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)
( 89,166)( 90,168)( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)( 96,180)
( 97,176)( 98,175)( 99,177)(100,182)(101,181)(102,183)(103,188)(104,187)
(105,189)(106,185)(107,184)(108,186);;
s2 := (  1,166)(  2,167)(  3,168)(  4,163)(  5,164)(  6,165)(  7,169)(  8,170)
(  9,171)( 10,184)( 11,185)( 12,186)( 13,181)( 14,182)( 15,183)( 16,187)
( 17,188)( 18,189)( 19,175)( 20,176)( 21,177)( 22,172)( 23,173)( 24,174)
( 25,178)( 26,179)( 27,180)( 28,193)( 29,194)( 30,195)( 31,190)( 32,191)
( 33,192)( 34,196)( 35,197)( 36,198)( 37,211)( 38,212)( 39,213)( 40,208)
( 41,209)( 42,210)( 43,214)( 44,215)( 45,216)( 46,202)( 47,203)( 48,204)
( 49,199)( 50,200)( 51,201)( 52,205)( 53,206)( 54,207)( 55,139)( 56,140)
( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)( 64,157)
( 65,158)( 66,159)( 67,154)( 68,155)( 69,156)( 70,160)( 71,161)( 72,162)
( 73,148)( 74,149)( 75,150)( 76,145)( 77,146)( 78,147)( 79,151)( 80,152)
( 81,153)( 82,112)( 83,113)( 84,114)( 85,109)( 86,110)( 87,111)( 88,115)
( 89,116)( 90,117)( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)( 96,129)
( 97,133)( 98,134)( 99,135)(100,121)(101,122)(102,123)(103,118)(104,119)
(105,120)(106,124)(107,125)(108,126);;
s3 := (  1,172)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)(  8,176)
(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)( 16,166)
( 17,167)( 18,168)( 19,181)( 20,182)( 21,183)( 22,187)( 23,188)( 24,189)
( 25,184)( 26,185)( 27,186)( 28,199)( 29,200)( 30,201)( 31,205)( 32,206)
( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)( 40,196)
( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)( 48,210)
( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,145)( 56,146)
( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)( 64,136)
( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)( 72,141)
( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)
( 81,159)( 82,118)( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)( 88,121)
( 89,122)( 90,123)( 91,109)( 92,110)( 93,111)( 94,115)( 95,116)( 96,117)
( 97,112)( 98,113)( 99,114)(100,127)(101,128)(102,129)(103,133)(104,134)
(105,135)(106,130)(107,131)(108,132);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(216)!(  2,  3)(  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)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 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)(109,136)(110,138)
(111,137)(112,139)(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)(164,165)(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);
s1 := Sym(216)!(  1,110)(  2,109)(  3,111)(  4,116)(  5,115)(  6,117)(  7,113)
(  8,112)(  9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)
( 16,122)( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)
( 24,135)( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)
( 32,142)( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)
( 40,152)( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)
( 48,156)( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,191)
( 56,190)( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)
( 64,200)( 65,199)( 66,201)( 67,206)( 68,205)( 69,207)( 70,203)( 71,202)
( 72,204)( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)
( 80,211)( 81,213)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)
( 88,167)( 89,166)( 90,168)( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)
( 96,180)( 97,176)( 98,175)( 99,177)(100,182)(101,181)(102,183)(103,188)
(104,187)(105,189)(106,185)(107,184)(108,186);
s2 := Sym(216)!(  1,166)(  2,167)(  3,168)(  4,163)(  5,164)(  6,165)(  7,169)
(  8,170)(  9,171)( 10,184)( 11,185)( 12,186)( 13,181)( 14,182)( 15,183)
( 16,187)( 17,188)( 18,189)( 19,175)( 20,176)( 21,177)( 22,172)( 23,173)
( 24,174)( 25,178)( 26,179)( 27,180)( 28,193)( 29,194)( 30,195)( 31,190)
( 32,191)( 33,192)( 34,196)( 35,197)( 36,198)( 37,211)( 38,212)( 39,213)
( 40,208)( 41,209)( 42,210)( 43,214)( 44,215)( 45,216)( 46,202)( 47,203)
( 48,204)( 49,199)( 50,200)( 51,201)( 52,205)( 53,206)( 54,207)( 55,139)
( 56,140)( 57,141)( 58,136)( 59,137)( 60,138)( 61,142)( 62,143)( 63,144)
( 64,157)( 65,158)( 66,159)( 67,154)( 68,155)( 69,156)( 70,160)( 71,161)
( 72,162)( 73,148)( 74,149)( 75,150)( 76,145)( 77,146)( 78,147)( 79,151)
( 80,152)( 81,153)( 82,112)( 83,113)( 84,114)( 85,109)( 86,110)( 87,111)
( 88,115)( 89,116)( 90,117)( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)
( 96,129)( 97,133)( 98,134)( 99,135)(100,121)(101,122)(102,123)(103,118)
(104,119)(105,120)(106,124)(107,125)(108,126);
s3 := Sym(216)!(  1,172)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)
(  8,176)(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)
( 16,166)( 17,167)( 18,168)( 19,181)( 20,182)( 21,183)( 22,187)( 23,188)
( 24,189)( 25,184)( 26,185)( 27,186)( 28,199)( 29,200)( 30,201)( 31,205)
( 32,206)( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)
( 40,196)( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)
( 48,210)( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,145)
( 56,146)( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)
( 64,136)( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)
( 72,141)( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)
( 80,158)( 81,159)( 82,118)( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)
( 88,121)( 89,122)( 90,123)( 91,109)( 92,110)( 93,111)( 94,115)( 95,116)
( 96,117)( 97,112)( 98,113)( 99,114)(100,127)(101,128)(102,129)(103,133)
(104,134)(105,135)(106,130)(107,131)(108,132);
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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope